<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">s1</font>, <font color="Maroon" title="c2">t1</font>, <font color="Maroon" title="c3">s2</font>, <font color="Maroon" title="c4">t2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">P1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:47"/><i><font color="Green" title="E46">A1</font></i>: 
( <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> )
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green" title="E47">A2</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">t1</font> ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">t1</font> ) ) </span>}</span>  &amp; <font color="Maroon" title="c6">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">pa</font> where <font color="Olive" title="b1">pa</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">pa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">pa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">pa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">pa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span>  )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">P2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">pb</font> where <font color="Olive" title="b1">pb</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">pb</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <font color="Olive" title="b1">pb</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> or  not <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">pb</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <font color="Olive" title="b1">pb</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan1.html#T35" target="_self" title="JORDAN1:th.35">Th35</a></i>;<br/>
<a NAME="E4:47"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">PP</font> = <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E6:47"/>
<font color="Maroon" title="c7">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">pb</font> where <font color="Olive" title="b1">pb</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">pb</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <font color="Olive" title="b1">pb</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> or  not <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">pb</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <font color="Olive" title="b1">pb</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 
;<br/>
<a NAME="E7:47"/><b>then </b><i><font color="Green" title="E49">A4</font></i>: 
( <font color="Maroon" title="c8">PP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P2</font> &amp; <font color="Maroon" title="c8">PP</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">P2</font> )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="jordan1.html#T41" target="_self" title="JORDAN1:th.41">Th41</a></i>;<br/>
<a NAME="E8:47"/><b>then </b><i><font color="Green" title="E50">A5</font></i>: 
( <font color="Maroon" title="c7">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>
<a NAME="E9:47"/>
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47"><i><font color="Green" title="E49">A4</font></i></a></i>;<br/>
<a NAME="E10:47"/><b>then </b><i><font color="Green" title="E51">A6</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">P1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">P2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T53" title="XBOOLE_1:th.53">XBOOLE_1:53</a></i>;<br/>
<a NAME="E11:47"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></i>;<br/>
<a NAME="E12:47"/><b>then </b><i><font color="Green" title="E52">A7</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E13:47"/><i><font color="Green" title="E53">A8</font></i>: 
<font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/><a NAME="E2:47_1_1"/><b>then </b>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> &amp; not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:47_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E12:47"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:47_1"/><b>then </b><i><font color="Green" title="E54">A9</font></i>: 
<font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font>
 ;<br/><a NAME="E2:47_1_2"/><b>then </b>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:47_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan1.html#E8:47"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="jordan1.html#E10:47"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:47_1"/><b>then </b>
<font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:47_1"/>
<font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E2:47_1"><i><font color="Green" title="E54">A9</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:47"/><i><font color="Green" title="E54">A10</font></i>: 
<font color="Maroon" title="c7">P2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="jordan1.html#T39" target="_self" title="JORDAN1:th.39">Th39</a></i>;<br/>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 

<br/>.= 
<font color="Maroon" title="c7">P2</font>

;<br/>
<a NAME="E16:47"/><b>then </b><i><font color="Green" title="E55">A11</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="jordan1.html#E13:47"><i><font color="Green" title="E53">A8</font></i></a>, <a class="txt" href="jordan1.html#E14:47"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E17:47"/><i><font color="Green" title="E56">A12</font></i>: 
<font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>thus </b><a NAME="E18:47"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 ;<br/>

<b>hence </b><a NAME="E2:47_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E16:47"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="jordan1.html#E17:47"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="pre_topc.html#T45" title="PRE_TOPC:th.45">PRE_TOPC:45</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E19:47"/>
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_4"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:47_4"/><i><font color="Green" title="E57">A13</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <b>and </b><br/><a NAME="E4:47_4"/><i><font color="Green" title="E58">A14</font></i>: 
( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">t1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">t1</font> ) )
 <b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c10">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_4_1_1"/>
( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">t1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">t1</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan1.html#E4:47_4"><i><font color="Green" title="E58">A14</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_1"><b>suppose </b></a><a NAME="E1:47_4_1_1_1"/><i><font color="Green" title="E59">A15</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">t1</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_4_1_1_1_1_1"/>
( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:47_4_1_1_1_1_1_1"/><i><font color="Green" title="E60">A16</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_1_1_1_1"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_1_1_1_1_1"/><i><font color="Green" title="E61">A17</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_1_1_1_1_1"/><i><font color="Green" title="E62">A18</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_1_1_1_1_1"/><i><font color="Green" title="E63">A19</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_1_1_1_1_1"/><i><font color="Green" title="E64">A20</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E61">A17</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E62">A18</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_1_1_1_1_1"/><i><font color="Green" title="E65">A21</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E63">A19</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E8:47_4_1_1_1_1_1_1_1"/><i><font color="Green" title="E66">A22</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E63">A19</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E9:47_4_1_1_1_1_1_1_1"/><i><font color="Green" title="E67">A23</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_1_1_1_1_1"/>
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E11:47_4_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E68">A24</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:47_4_1_1_1_1_1_1_1"/><i><font color="Green" title="E69">A25</font></i>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E66">A22</font></i></a>, <a class="txt" href="jordan1.html#E9:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:47_4_1_1_1_1_1_1_1"/><i><font color="Green" title="E70">A26</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E14:47_4_1_1_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E15:47_4_1_1_1_1_1_1_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E16:47_4_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E71">A27</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E65">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E17:47_4_1_1_1_1_1_1_1"/><i><font color="Green" title="E72">A28</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E18:47_4_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E13:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E70">A26</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_1_1_1_1_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_1"><i><font color="Green" title="E60">A16</font></i></a>, <a class="txt" href="jordan1.html#E11:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E68">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_1"><i><font color="Green" title="E60">A16</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E71">A27</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E72">A28</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E21:47_4_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E73">A29</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E23:47_4_1_1_1_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E24:47_4_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E74">A30</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E25:47_4_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
;<br/>
<a NAME="E26:47_4_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E12:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E69">A25</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E71">A27</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E72">A28</font></i></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/>
<a NAME="E27:47_4_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E24:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E74">A30</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E28:47_4_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E75">A31</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E63">A19</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E29:47_4_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E30:47_4_1_1_1_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E28:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E75">A31</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E31:47_4_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E32:47_4_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E64">A20</font></i></a>, <a class="txt" href="jordan1.html#E21:47_4_1_1_1_1_1_1_1"><i><font color="Green" title="E73">A29</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:47_4_1_1_1_1_1_2"/><i><font color="Green" title="E60">A32</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_1_1_1_2"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E61">A33</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E62">A34</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E63">A35</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E64">A36</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E61">A33</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E62">A34</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E65">A37</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E63">A35</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E8:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E66">A38</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E67">A39</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E68">A40</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E66">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E69">A41</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E12:47_4_1_1_1_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E13:47_4_1_1_1_1_1_2_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E14:47_4_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E70">A42</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E65">A37</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E15:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E71">A43</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E16:47_4_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E72">A44</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_2"><i><font color="Green" title="E60">A32</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E70">A42</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E17:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E73">A45</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_2"><i><font color="Green" title="E60">A32</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E70">A42</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E71">A43</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E18:47_4_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E69">A41</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_1_1_1_2_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_2"><i><font color="Green" title="E60">A32</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E68">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E74">A46</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E71">A43</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E21:47_4_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E69">A41</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E22:47_4_1_1_1_1_1_2_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_2"><i><font color="Green" title="E60">A32</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E68">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:47_4_1_1_1_1_1_2_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E71">A43</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E24:47_4_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E75">A47</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E72">A44</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E73">A45</font></i></a>, <a class="txt" href="jordan1.html#E20:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E74">A46</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E26:47_4_1_1_1_1_1_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E27:47_4_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E76">A48</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E28:47_4_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
;<br/>
<a NAME="E29:47_4_1_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E67">A39</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E70">A42</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E71">A43</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E30:47_4_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E77">A49</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E71">A43</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E31:47_4_1_1_1_1_1_2_1"/><i><font color="Green" title="E78">A50</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E32:47_4_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E63">A35</font></i></a>, <a class="txt" href="jordan1.html#E7:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E65">A37</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E33:47_4_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E31:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E78">A50</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E34:47_4_1_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E30:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E77">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:47_4_1_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E27:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E76">A48</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E36:47_4_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E79">A51</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E63">A35</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E37:47_4_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E38:47_4_1_1_1_1_1_2_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E36:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E79">A51</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E39:47_4_1_1_1_1_1_2_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E40:47_4_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="jordan1.html#E24:47_4_1_1_1_1_1_2_1"><i><font color="Green" title="E75">A47</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:47_4_1_1_1_1_1_3"/><i><font color="Green" title="E60">A52</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_1_1_1_3"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E61">A53</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E62">A54</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E63">A55</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E64">A56</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E61">A53</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E62">A54</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E65">A57</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E63">A55</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E8:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E66">A58</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E67">A59</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E68">A60</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E66">A58</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E69">A61</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E12:47_4_1_1_1_1_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E13:47_4_1_1_1_1_1_3_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E14:47_4_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E70">A62</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E65">A57</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E15:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E71">A63</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E16:47_4_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E72">A64</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_3"><i><font color="Green" title="E60">A52</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E70">A62</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E17:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E73">A65</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_3"><i><font color="Green" title="E60">A52</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E70">A62</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E71">A63</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E18:47_4_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E69">A61</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_1_1_1_3_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_3"><i><font color="Green" title="E60">A52</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E68">A60</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E74">A66</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E71">A63</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E21:47_4_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E69">A61</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E22:47_4_1_1_1_1_1_3_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_1_1_1_3"><i><font color="Green" title="E60">A52</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E68">A60</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:47_4_1_1_1_1_1_3_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E71">A63</font></i></a>, <a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a></i>;<br/>
<a NAME="E24:47_4_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E75">A67</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E72">A64</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E73">A65</font></i></a>, <a class="txt" href="jordan1.html#E20:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E74">A66</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E26:47_4_1_1_1_1_1_3_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E27:47_4_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E76">A68</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E28:47_4_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E67">A59</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E70">A62</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E29:47_4_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E77">A69</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E71">A63</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E30:47_4_1_1_1_1_1_3_1"/><i><font color="Green" title="E78">A70</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E31:47_4_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E63">A55</font></i></a>, <a class="txt" href="jordan1.html#E7:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E65">A57</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E32:47_4_1_1_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E30:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E78">A70</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E33:47_4_1_1_1_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E29:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E77">A69</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:47_4_1_1_1_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E27:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E76">A68</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E35:47_4_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E79">A71</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E63">A55</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E36:47_4_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E37:47_4_1_1_1_1_1_3_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E35:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E79">A71</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E38:47_4_1_1_1_1_1_3_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E39:47_4_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E64">A56</font></i></a>, <a class="txt" href="jordan1.html#E24:47_4_1_1_1_1_1_3_1"><i><font color="Green" title="E75">A67</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_1_1_1_3"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E3:47_4"><i><font color="Green" title="E57">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_2"><b>suppose </b></a><a NAME="E1:47_4_1_1_2"/><i><font color="Green" title="E59">A72</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_4_1_1_2_1_1"/>
( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2"><i><font color="Green" title="E59">A72</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:47_4_1_1_2_1_1_1"/><i><font color="Green" title="E60">A73</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_2_1_1_1"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_2_1_1_1_1"/><i><font color="Green" title="E61">A74</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_2_1_1_1_1"/><i><font color="Green" title="E62">A75</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_2_1_1_1_1"/><i><font color="Green" title="E63">A76</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_2_1_1_1_1"/><i><font color="Green" title="E64">A77</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E61">A74</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E62">A75</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_2_1_1_1_1"/><i><font color="Green" title="E65">A78</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E8:47_4_1_1_2_1_1_1_1"/><i><font color="Green" title="E66">A79</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E9:47_4_1_1_2_1_1_1_1"/><i><font color="Green" title="E67">A80</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_2_1_1_1_1"/>
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E11:47_4_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E68">A81</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E67">A80</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:47_4_1_1_2_1_1_1_1"/><i><font color="Green" title="E69">A82</font></i>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E66">A79</font></i></a>, <a class="txt" href="jordan1.html#E9:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E67">A80</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:47_4_1_1_2_1_1_1_1"/><i><font color="Green" title="E70">A83</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E14:47_4_1_1_2_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E15:47_4_1_1_2_1_1_1_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E16:47_4_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E71">A84</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E65">A78</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E17:47_4_1_1_2_1_1_1_1"/><i><font color="Green" title="E72">A85</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E18:47_4_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E13:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E70">A83</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_2_1_1_1_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_1"><i><font color="Green" title="E60">A73</font></i></a>, <a class="txt" href="jordan1.html#E11:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E68">A81</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_1"><i><font color="Green" title="E60">A73</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E71">A84</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E72">A85</font></i></a>, <a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E21:47_4_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E73">A86</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E23:47_4_1_1_2_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E24:47_4_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E74">A87</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E25:47_4_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E12:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E69">A82</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E71">A84</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E72">A85</font></i></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/>
<a NAME="E26:47_4_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E24:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E74">A87</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E27:47_4_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E75">A88</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E28:47_4_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E29:47_4_1_1_2_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E27:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E75">A88</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E30:47_4_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E31:47_4_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="jordan1.html#E21:47_4_1_1_2_1_1_1_1"><i><font color="Green" title="E73">A86</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:47_4_1_1_2_1_1_2"/><i><font color="Green" title="E60">A89</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_2_1_1_2"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E61">A90</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E62">A91</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E63">A92</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E64">A93</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E61">A90</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E62">A91</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E65">A94</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E63">A92</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E8:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E66">A95</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E67">A96</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E68">A97</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E66">A95</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E69">A98</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E12:47_4_1_1_2_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E13:47_4_1_1_2_1_1_2_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E14:47_4_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E70">A99</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E65">A94</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E15:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E71">A100</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E16:47_4_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E72">A101</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_2"><i><font color="Green" title="E60">A89</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E70">A99</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E17:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E73">A102</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_2"><i><font color="Green" title="E60">A89</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E70">A99</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E71">A100</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E18:47_4_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E69">A98</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_2_1_1_2_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_2"><i><font color="Green" title="E60">A89</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E68">A97</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E74">A103</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E71">A100</font></i></a>, <a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a></i>;<br/>
<a NAME="E21:47_4_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E69">A98</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E22:47_4_1_1_2_1_1_2_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_2"><i><font color="Green" title="E60">A89</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E68">A97</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:47_4_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E71">A100</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E24:47_4_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E75">A104</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E72">A101</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E73">A102</font></i></a>, <a class="txt" href="jordan1.html#E20:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E74">A103</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E26:47_4_1_1_2_1_1_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E27:47_4_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E76">A105</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E28:47_4_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E67">A96</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E70">A99</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E71">A100</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E29:47_4_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E77">A106</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E71">A100</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E30:47_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E78">A107</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E31:47_4_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E63">A92</font></i></a>, <a class="txt" href="jordan1.html#E7:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E65">A94</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E32:47_4_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E30:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E78">A107</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E33:47_4_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E29:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E77">A106</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:47_4_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E27:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E76">A105</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E35:47_4_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E79">A108</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E63">A92</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E36:47_4_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E37:47_4_1_1_2_1_1_2_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E35:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E79">A108</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E38:47_4_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E39:47_4_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E64">A93</font></i></a>, <a class="txt" href="jordan1.html#E24:47_4_1_1_2_1_1_2_1"><i><font color="Green" title="E75">A104</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:47_4_1_1_2_1_1_3"/><i><font color="Green" title="E60">A109</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_2_1_1_3"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_2_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E61">A110</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E62">A111</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E63">A112</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E64">A113</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E61">A110</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E62">A111</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E65">A114</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E63">A112</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E8:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E66">A115</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E67">A116</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E68">A117</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E66">A115</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E69">A118</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E12:47_4_1_1_2_1_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E13:47_4_1_1_2_1_1_3_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E14:47_4_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E70">A119</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E65">A114</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E15:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E71">A120</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E16:47_4_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E72">A121</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_3"><i><font color="Green" title="E60">A109</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E70">A119</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E17:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E73">A122</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_3"><i><font color="Green" title="E60">A109</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E70">A119</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E71">A120</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E18:47_4_1_1_2_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E69">A118</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_2_1_1_3_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_3"><i><font color="Green" title="E60">A109</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E68">A117</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E74">A123</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E71">A120</font></i></a>, <a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a></i>;<br/>
<a NAME="E21:47_4_1_1_2_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E69">A118</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E22:47_4_1_1_2_1_1_3_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_2_1_1_3"><i><font color="Green" title="E60">A109</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E68">A117</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:47_4_1_1_2_1_1_3_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E71">A120</font></i></a>, <a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a></i>;<br/>
<a NAME="E24:47_4_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E75">A124</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E72">A121</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E73">A122</font></i></a>, <a class="txt" href="jordan1.html#E20:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E74">A123</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E26:47_4_1_1_2_1_1_3_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E27:47_4_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E76">A125</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E28:47_4_1_1_2_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E67">A116</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E70">A119</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E29:47_4_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E77">A126</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E71">A120</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E30:47_4_1_1_2_1_1_3_1"/><i><font color="Green" title="E78">A127</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E31:47_4_1_1_2_1_1_3_1"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E63">A112</font></i></a>, <a class="txt" href="jordan1.html#E7:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E65">A114</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E32:47_4_1_1_2_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E30:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E78">A127</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E33:47_4_1_1_2_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E29:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E77">A126</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:47_4_1_1_2_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E27:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E76">A125</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E35:47_4_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E79">A128</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E63">A112</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E36:47_4_1_1_2_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E37:47_4_1_1_2_1_1_3_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E35:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E79">A128</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E38:47_4_1_1_2_1_1_3_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E39:47_4_1_1_2_1_1_3_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E64">A113</font></i></a>, <a class="txt" href="jordan1.html#E24:47_4_1_1_2_1_1_3_1"><i><font color="Green" title="E75">A124</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_2_1_1_3"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E3:47_4"><i><font color="Green" title="E57">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_3"><b>suppose </b></a><a NAME="E1:47_4_1_1_3"/><i><font color="Green" title="E59">A129</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">s1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_4_1_1_3_1_1"/>
( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3"><i><font color="Green" title="E59">A129</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:47_4_1_1_3_1_1_1"/><i><font color="Green" title="E60">A130</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_3_1_1_1"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_3_1_1_1_1"/><i><font color="Green" title="E61">A131</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_3_1_1_1_1"/><i><font color="Green" title="E62">A132</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_3_1_1_1_1"/><i><font color="Green" title="E63">A133</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_3_1_1_1_1"/><i><font color="Green" title="E64">A134</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E61">A131</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E62">A132</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_3_1_1_1_1"/><i><font color="Green" title="E65">A135</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E63">A133</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E8:47_4_1_1_3_1_1_1_1"/><i><font color="Green" title="E66">A136</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E63">A133</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E9:47_4_1_1_3_1_1_1_1"/><i><font color="Green" title="E67">A137</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_3_1_1_1_1"/>
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E11:47_4_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E68">A138</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E67">A137</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:47_4_1_1_3_1_1_1_1"/><i><font color="Green" title="E69">A139</font></i>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E66">A136</font></i></a>, <a class="txt" href="jordan1.html#E9:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E67">A137</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:47_4_1_1_3_1_1_1_1"/><i><font color="Green" title="E70">A140</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E14:47_4_1_1_3_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E15:47_4_1_1_3_1_1_1_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E16:47_4_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E71">A141</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E65">A135</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E17:47_4_1_1_3_1_1_1_1"/><i><font color="Green" title="E72">A142</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E18:47_4_1_1_3_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E13:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E70">A140</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_3_1_1_1_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_1"><i><font color="Green" title="E60">A130</font></i></a>, <a class="txt" href="jordan1.html#E11:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E68">A138</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_3_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_1"><i><font color="Green" title="E60">A130</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E71">A141</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E72">A142</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E21:47_4_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E73">A143</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E23:47_4_1_1_3_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E24:47_4_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E74">A144</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E25:47_4_1_1_3_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
;<br/>
<a NAME="E26:47_4_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E12:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E69">A139</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E71">A141</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E72">A142</font></i></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/>
<a NAME="E27:47_4_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E24:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E74">A144</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E28:47_4_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E75">A145</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E63">A133</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E29:47_4_1_1_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E30:47_4_1_1_3_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E28:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E75">A145</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E31:47_4_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E32:47_4_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E64">A134</font></i></a>, <a class="txt" href="jordan1.html#E21:47_4_1_1_3_1_1_1_1"><i><font color="Green" title="E73">A143</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_3_1_1_1"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:47_4_1_1_3_1_1_2"/><i><font color="Green" title="E60">A146</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s1</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_3_1_1_2"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E61">A147</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E62">A148</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E63">A149</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E64">A150</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E61">A147</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E62">A148</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E65">A151</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E63">A149</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E8:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E66">A152</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E67">A153</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_3_1_1_2_1"/><b>then </b><i><font color="Green" title="E68">A154</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E66">A152</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E69">A155</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E12:47_4_1_1_3_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E13:47_4_1_1_3_1_1_2_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E14:47_4_1_1_3_1_1_2_1"/><b>then </b><i><font color="Green" title="E70">A156</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E65">A151</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E15:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E71">A157</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E16:47_4_1_1_3_1_1_2_1"/><b>then </b><i><font color="Green" title="E72">A158</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_2"><i><font color="Green" title="E60">A146</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E70">A156</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E17:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E73">A159</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_2"><i><font color="Green" title="E60">A146</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E70">A156</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E71">A157</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E18:47_4_1_1_3_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E69">A155</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_3_1_1_2_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_2"><i><font color="Green" title="E60">A146</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E68">A154</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_3_1_1_2_1"/><b>then </b><i><font color="Green" title="E74">A160</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E71">A157</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E21:47_4_1_1_3_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E69">A155</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E22:47_4_1_1_3_1_1_2_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_2"><i><font color="Green" title="E60">A146</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E68">A154</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:47_4_1_1_3_1_1_2_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E71">A157</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E24:47_4_1_1_3_1_1_2_1"/><b>then </b><i><font color="Green" title="E75">A161</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E72">A158</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E73">A159</font></i></a>, <a class="txt" href="jordan1.html#E20:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E74">A160</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E26:47_4_1_1_3_1_1_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E27:47_4_1_1_3_1_1_2_1"/><b>then </b><i><font color="Green" title="E76">A162</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E28:47_4_1_1_3_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
;<br/>
<a NAME="E29:47_4_1_1_3_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E67">A153</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E70">A156</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E71">A157</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E30:47_4_1_1_3_1_1_2_1"/><b>then </b><i><font color="Green" title="E77">A163</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E71">A157</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E31:47_4_1_1_3_1_1_2_1"/><i><font color="Green" title="E78">A164</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E32:47_4_1_1_3_1_1_2_1"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E63">A149</font></i></a>, <a class="txt" href="jordan1.html#E7:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E65">A151</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E33:47_4_1_1_3_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E31:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E78">A164</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E34:47_4_1_1_3_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E30:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E77">A163</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:47_4_1_1_3_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E27:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E76">A162</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E36:47_4_1_1_3_1_1_2_1"/><b>then </b><i><font color="Green" title="E79">A165</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E63">A149</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E37:47_4_1_1_3_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E38:47_4_1_1_3_1_1_2_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E36:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E79">A165</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E39:47_4_1_1_3_1_1_2_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E40:47_4_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E64">A150</font></i></a>, <a class="txt" href="jordan1.html#E24:47_4_1_1_3_1_1_2_1"><i><font color="Green" title="E75">A161</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_3_1_1_2"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:47_4_1_1_3_1_1_3"/><i><font color="Green" title="E60">A166</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_3_1_1_3"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_3_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E61">A167</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E62">A168</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E63">A169</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E64">A170</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E61">A167</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E62">A168</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E65">A171</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E63">A169</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E8:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E66">A172</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E67">A173</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_3_1_1_3_1"/><b>then </b><i><font color="Green" title="E68">A174</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E66">A172</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E69">A175</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E12:47_4_1_1_3_1_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E13:47_4_1_1_3_1_1_3_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E14:47_4_1_1_3_1_1_3_1"/><b>then </b><i><font color="Green" title="E70">A176</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E65">A171</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E15:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E71">A177</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E16:47_4_1_1_3_1_1_3_1"/><b>then </b><i><font color="Green" title="E72">A178</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_3"><i><font color="Green" title="E60">A166</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E70">A176</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E17:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E73">A179</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_3"><i><font color="Green" title="E60">A166</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E70">A176</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E71">A177</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E18:47_4_1_1_3_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E69">A175</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_3_1_1_3_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_3"><i><font color="Green" title="E60">A166</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E68">A174</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_3_1_1_3_1"/><b>then </b><i><font color="Green" title="E74">A180</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E71">A177</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E21:47_4_1_1_3_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E69">A175</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E22:47_4_1_1_3_1_1_3_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_3_1_1_3"><i><font color="Green" title="E60">A166</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E68">A174</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:47_4_1_1_3_1_1_3_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E71">A177</font></i></a>, <a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a></i>;<br/>
<a NAME="E24:47_4_1_1_3_1_1_3_1"/><b>then </b><i><font color="Green" title="E75">A181</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E72">A178</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E73">A179</font></i></a>, <a class="txt" href="jordan1.html#E20:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E74">A180</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E26:47_4_1_1_3_1_1_3_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E27:47_4_1_1_3_1_1_3_1"/><b>then </b><i><font color="Green" title="E76">A182</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E28:47_4_1_1_3_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E67">A173</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E70">A176</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E29:47_4_1_1_3_1_1_3_1"/><b>then </b><i><font color="Green" title="E77">A183</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E71">A177</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E30:47_4_1_1_3_1_1_3_1"/><i><font color="Green" title="E78">A184</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E31:47_4_1_1_3_1_1_3_1"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E63">A169</font></i></a>, <a class="txt" href="jordan1.html#E7:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E65">A171</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E32:47_4_1_1_3_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E30:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E78">A184</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E33:47_4_1_1_3_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E29:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E77">A183</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:47_4_1_1_3_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E27:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E76">A182</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E35:47_4_1_1_3_1_1_3_1"/><b>then </b><i><font color="Green" title="E79">A185</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E63">A169</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E36:47_4_1_1_3_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E37:47_4_1_1_3_1_1_3_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E35:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E79">A185</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E38:47_4_1_1_3_1_1_3_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E39:47_4_1_1_3_1_1_3_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E64">A170</font></i></a>, <a class="txt" href="jordan1.html#E24:47_4_1_1_3_1_1_3_1"><i><font color="Green" title="E75">A181</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_3_1_1_3"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E3:47_4"><i><font color="Green" title="E57">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_4"><b>suppose </b></a><a NAME="E1:47_4_1_1_4"/><i><font color="Green" title="E59">A186</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">t1</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_4_1_1_4_1_1"/>
( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4"><i><font color="Green" title="E59">A186</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:47_4_1_1_4_1_1_1"/><i><font color="Green" title="E60">A187</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_4_1_1_1"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_4_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_4_1_1_1_1"/><i><font color="Green" title="E61">A188</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_4_1_1_1_1"/><i><font color="Green" title="E62">A189</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_4_1_1_1_1"/><i><font color="Green" title="E63">A190</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_4_1_1_1_1"/><i><font color="Green" title="E64">A191</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E61">A188</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E62">A189</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_4_1_1_1_1"/><i><font color="Green" title="E65">A192</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E63">A190</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E8:47_4_1_1_4_1_1_1_1"/><i><font color="Green" title="E66">A193</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E63">A190</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E9:47_4_1_1_4_1_1_1_1"/><i><font color="Green" title="E67">A194</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_4_1_1_1_1"/>
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E11:47_4_1_1_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E68">A195</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E67">A194</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:47_4_1_1_4_1_1_1_1"/><i><font color="Green" title="E69">A196</font></i>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E66">A193</font></i></a>, <a class="txt" href="jordan1.html#E9:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E67">A194</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:47_4_1_1_4_1_1_1_1"/><i><font color="Green" title="E70">A197</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E14:47_4_1_1_4_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E15:47_4_1_1_4_1_1_1_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E16:47_4_1_1_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E71">A198</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E65">A192</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E17:47_4_1_1_4_1_1_1_1"/><i><font color="Green" title="E72">A199</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E18:47_4_1_1_4_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E13:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E70">A197</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_4_1_1_1_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_1"><i><font color="Green" title="E60">A187</font></i></a>, <a class="txt" href="jordan1.html#E11:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E68">A195</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_4_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_1"><i><font color="Green" title="E60">A187</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E71">A198</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E72">A199</font></i></a>, <a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E21:47_4_1_1_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E73">A200</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E23:47_4_1_1_4_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E24:47_4_1_1_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E74">A201</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E25:47_4_1_1_4_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E12:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E69">A196</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E71">A198</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E72">A199</font></i></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/>
<a NAME="E26:47_4_1_1_4_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E24:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E74">A201</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E27:47_4_1_1_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E75">A202</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E63">A190</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E28:47_4_1_1_4_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E29:47_4_1_1_4_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E27:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E75">A202</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E30:47_4_1_1_4_1_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E31:47_4_1_1_4_1_1_1_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E64">A191</font></i></a>, <a class="txt" href="jordan1.html#E21:47_4_1_1_4_1_1_1_1"><i><font color="Green" title="E73">A200</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_4_1_1_1"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:47_4_1_1_4_1_1_2"/><i><font color="Green" title="E60">A203</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t1</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_4_1_1_2"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_4_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E61">A204</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E62">A205</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E63">A206</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E64">A207</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E61">A204</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E62">A205</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E65">A208</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E63">A206</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E8:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E66">A209</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E67">A210</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_4_1_1_2_1"/><b>then </b><i><font color="Green" title="E68">A211</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E66">A209</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E69">A212</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E12:47_4_1_1_4_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E13:47_4_1_1_4_1_1_2_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E14:47_4_1_1_4_1_1_2_1"/><b>then </b><i><font color="Green" title="E70">A213</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E65">A208</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E15:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E71">A214</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E16:47_4_1_1_4_1_1_2_1"/><b>then </b><i><font color="Green" title="E72">A215</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_2"><i><font color="Green" title="E60">A203</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E70">A213</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E17:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E73">A216</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_2"><i><font color="Green" title="E60">A203</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E70">A213</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E71">A214</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E18:47_4_1_1_4_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E69">A212</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_4_1_1_2_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_2"><i><font color="Green" title="E60">A203</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E68">A211</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_4_1_1_2_1"/><b>then </b><i><font color="Green" title="E74">A217</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E71">A214</font></i></a>, <a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a></i>;<br/>
<a NAME="E21:47_4_1_1_4_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E69">A212</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E22:47_4_1_1_4_1_1_2_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_2"><i><font color="Green" title="E60">A203</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E68">A211</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:47_4_1_1_4_1_1_2_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E71">A214</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E24:47_4_1_1_4_1_1_2_1"/><b>then </b><i><font color="Green" title="E75">A218</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E72">A215</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E73">A216</font></i></a>, <a class="txt" href="jordan1.html#E20:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E74">A217</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E26:47_4_1_1_4_1_1_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E27:47_4_1_1_4_1_1_2_1"/><b>then </b><i><font color="Green" title="E76">A219</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E28:47_4_1_1_4_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E67">A210</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E70">A213</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E71">A214</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E29:47_4_1_1_4_1_1_2_1"/><b>then </b><i><font color="Green" title="E77">A220</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E71">A214</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E30:47_4_1_1_4_1_1_2_1"/><i><font color="Green" title="E78">A221</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E31:47_4_1_1_4_1_1_2_1"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E63">A206</font></i></a>, <a class="txt" href="jordan1.html#E7:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E65">A208</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E32:47_4_1_1_4_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E30:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E78">A221</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E33:47_4_1_1_4_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E29:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E77">A220</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:47_4_1_1_4_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E27:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E76">A219</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E35:47_4_1_1_4_1_1_2_1"/><b>then </b><i><font color="Green" title="E79">A222</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E63">A206</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E36:47_4_1_1_4_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E37:47_4_1_1_4_1_1_2_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E35:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E79">A222</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E38:47_4_1_1_4_1_1_2_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E39:47_4_1_1_4_1_1_2_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E64">A207</font></i></a>, <a class="txt" href="jordan1.html#E24:47_4_1_1_4_1_1_2_1"><i><font color="Green" title="E75">A218</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_4_1_1_2"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_4_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:47_4_1_1_4_1_1_3"/><i><font color="Green" title="E60">A223</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t2</font> )
 ;<br/><div class="add"><a NAME="E2:47_4_1_1_4_1_1_3"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1_1_4_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E61">A224</font></i>: 
<font color="Maroon" title="c12">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E62">A225</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E63">A226</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E64">A227</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E61">A224</font></i></a>, <a class="txt" href="jordan1.html#E2:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E62">A225</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E65">A228</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E63">A226</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">r2</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E8:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E66">A229</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E67">A230</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:47_4_1_1_4_1_1_3_1"/><b>then </b><i><font color="Green" title="E68">A231</font></i>: 
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E66">A229</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E69">A232</font></i>: 
( <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E12:47_4_1_1_4_1_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<a NAME="E13:47_4_1_1_4_1_1_3_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E14:47_4_1_1_4_1_1_3_1"/><b>then </b><i><font color="Green" title="E70">A233</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E7:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E65">A228</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">pa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E15:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E71">A234</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E16:47_4_1_1_4_1_1_3_1"/><b>then </b><i><font color="Green" title="E72">A235</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_3"><i><font color="Green" title="E60">A223</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E70">A233</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E17:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E73">A236</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_3"><i><font color="Green" title="E60">A223</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E70">A233</font></i></a>, <a class="txt" href="jordan1.html#E15:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E71">A234</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E18:47_4_1_1_4_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E69">A232</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E19:47_4_1_1_4_1_1_3_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_3"><i><font color="Green" title="E60">A223</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E68">A231</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:47_4_1_1_4_1_1_3_1"/><b>then </b><i><font color="Green" title="E74">A237</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E71">A234</font></i></a>, <a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a></i>;<br/>
<a NAME="E21:47_4_1_1_4_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E69">A232</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E22:47_4_1_1_4_1_1_3_1"/><b>then </b>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:47_4_1_1_4_1_1_3"><i><font color="Green" title="E60">A223</font></i></a>, <a class="txt" href="jordan1.html#E10:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E68">A231</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:47_4_1_1_4_1_1_3_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">t1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E71">A234</font></i></a>, <a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a></i>;<br/>
<a NAME="E24:47_4_1_1_4_1_1_3_1"/><b>then </b><i><font color="Green" title="E75">A238</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="jordan1.html#E16:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E72">A235</font></i></a>, <a class="txt" href="jordan1.html#E17:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E73">A236</font></i></a>, <a class="txt" href="jordan1.html#E20:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E74">A237</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E26:47_4_1_1_4_1_1_3_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E27:47_4_1_1_4_1_1_3_1"/><b>then </b><i><font color="Green" title="E76">A239</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E28:47_4_1_1_4_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E9:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E67">A230</font></i></a>, <a class="txt" href="jordan1.html#E14:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E70">A233</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E29:47_4_1_1_4_1_1_3_1"/><b>then </b><i><font color="Green" title="E77">A240</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E71">A234</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E30:47_4_1_1_4_1_1_3_1"/><i><font color="Green" title="E78">A241</font></i>: 
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E31:47_4_1_1_4_1_1_3_1"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E63">A226</font></i></a>, <a class="txt" href="jordan1.html#E7:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E65">A228</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E32:47_4_1_1_4_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1.html#E30:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E78">A241</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E33:47_4_1_1_4_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E29:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E77">A240</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:47_4_1_1_4_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E27:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E76">A239</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E35:47_4_1_1_4_1_1_3_1"/><b>then </b><i><font color="Green" title="E79">A242</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E63">A226</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E36:47_4_1_1_4_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> 2</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E37:47_4_1_1_4_1_1_3_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c17">qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E35:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E79">A242</font></i></a>, <a class="ref" href="topreal3.html#T12" title="TOPREAL3:th.12">TOPREAL3:12</a></i>;<br/>
<a NAME="E38:47_4_1_1_4_1_1_3_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">t2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c14">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E39:47_4_1_1_4_1_1_3_1"/>
<font color="Maroon" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E5:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E64">A227</font></i></a>, <a class="txt" href="jordan1.html#E24:47_4_1_1_4_1_1_3_1"><i><font color="Green" title="E75">A238</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_4_1_1_3"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:47_4_1_1_4"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E3:47_4"><i><font color="Green" title="E57">A13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:47_4"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:47"/>
<font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E4:47"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>


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