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

<b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">K0</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">Kb</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:59"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">K0</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span>  &amp; <font color="Maroon">Kb</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span>  )
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E4:59"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">D</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span><b> such that </b><br/><a NAME="E6:59"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  &amp; <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i/>;<br/>
<a NAME="E7:59"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">K0a</font> = <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">K1a</font> = <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span> ;<br/>
<a NAME="E8:59"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  &amp; <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  )
 <b>and </b><br/><a NAME="E2:59_1"/><i><font color="Green">E46</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E3:59_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:59_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">x1</font>, <font color="Maroon">p2</font> = <font color="Maroon">x2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jgraph_2.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<i><font color="Green">E48</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">K01</font> = <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E8:59_1"/>
( ( ( <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  ) or ( <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp;  <a href="jordan2c.html#K5">1.REAL</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 
<b>by </b><i>, <a class="ref" href="revrot_1.html#T19">REVROT_1:19</a></i>;<br/>
<a NAME="E9:59_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="jordan2c.html#K5">1.REAL</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span> 
 
;<br/>
<a NAME="E10:59_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:59_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p8</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p8</font> &amp; ( ( <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Maroon">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 ;<br/>
<a NAME="E4:59_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:59_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">K11</font> = <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E12:59_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K01</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">K11</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:59_1_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:59_1_3"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">px</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:59_1_3"/>
( ( ( ( <font color="Maroon">px</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">px</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">px</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">px</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">px</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">px</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">px</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">px</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">px</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) or ( ( ( <font color="Maroon">px</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">px</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">px</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">px</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">px</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">px</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">px</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">px</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Maroon">px</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) )
 
<b>by </b><i>, <a class="ref" href="real_2.html#T109">REAL_2:109</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:59_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K01</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K11</font> )
 
;<br/>
<b>hence </b><a NAME="E6:59_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K01</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">K11</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:59_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="jgraph_2.html#T10" target="_self">Th10</a>, , <a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K01</font> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K01</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p7</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_1_4_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p7</font> &amp; ( ( <font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p7</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 ;<br/><a NAME="E4:59_1_4_1_1"/><i><font color="Green">E58</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_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="E1:59_1_4_1_1_1_1"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  or ( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  &amp; not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  ) )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p8</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_1_4_1_1_1_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p8</font> &amp; ( ( <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 ;<br/><a NAME="E4:59_1_4_1_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, </i>;<br/><b>set </b><font color="Maroon">qq</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E5:59_1_4_1_1_1_1_1"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E6:59_1_4_1_1_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
( 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><i><font color="Green">E67</font></i>: <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K2">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 

;<br/><i><font color="Green">E68</font></i>: <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K2">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 

;<br/><div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_1_1_1_1_3"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:59_1_4_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:59_1_4_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/></div><b>end;</b></div><a NAME="E10:59_1_4_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T53">XCMPLX_1:53</a></i>;<br/><a NAME="E11:59_1_4_1_1_1_1_1"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xcmplx_1.html#T53">XCMPLX_1:53</a></i>;<br/><a NAME="E12:59_1_4_1_1_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E13:59_1_4_1_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  &amp; not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p8</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_1_4_1_1_1_1_2"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p8</font> &amp; ( ( <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Maroon">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 ;<br/><a NAME="E4:59_1_4_1_1_1_1_2"/>
( ( not <!--HUMANRECFAILED-->( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) or  not <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E5:59_1_4_1_1_1_1_2"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, </i>;<br/><a NAME="E6:59_1_4_1_1_1_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
( 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i>, , <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/><div><i><font color="Green">E152</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_1_1_1_2_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:59_1_4_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E3:59_1_4_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E154</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_1_1_1_2_2"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:59_1_4_1_1_1_1_2_2"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:59_1_4_1_1_1_1_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/></div><b>end;</b></div><i><font color="Green">E156</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/><i><font color="Green">E157</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/><a NAME="E11:59_1_4_1_1_1_1_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xcmplx_1.html#T113">XCMPLX_1:113</a></i>;<br/><a NAME="E12:59_1_4_1_1_1_1_2"/><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="xcmplx_1.html#T113">XCMPLX_1:113</a></i>;<br/><a NAME="E13:59_1_4_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E14:59_1_4_1_1_1_1_2"/><b>then </b><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E15:59_1_4_1_1_1_1_2"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E16:59_1_4_1_1_1_1_2"/><b>then </b><i><font color="Green">E162</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><b>consider </b><font color="Maroon">p9</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E18:59_1_4_1_1_1_1_2"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p9</font> &amp; ( ( <font color="Maroon">p9</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p9</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p9</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p9</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Maroon">p9</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a></i>;<br/><div><i><font color="Green">E164</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1_1_1_1_2_5_1"/>
( ( <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_5_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2_5_1_1"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_1_1_1_2_5_1_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:59_1_4_1_1_1_1_2_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, , <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_1_1_1_2_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_5_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2_5_1_2"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_5_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_1_1_1_2_5_1_2_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:59_1_4_1_1_1_1_2_5_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:59_1_4_1_1_1_1_2_5_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:59_1_4_1_1_1_1_2_5_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E3:59_1_4_1_1_1_1_2_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, , <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_1_1_1_2_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green">E168</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1_1_1_1_2_6_1"/>
( ( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_6_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2_6_1_1"/><i><font color="Green">E169</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_1_1_1_2_6_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:59_1_4_1_1_1_1_2_6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, , <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_1_1_1_2_6_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_6_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2_6_1_2"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_6_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_1_1_1_2_6_1_2_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:59_1_4_1_1_1_1_2_6_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:59_1_4_1_1_1_1_2_6_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:59_1_4_1_1_1_1_2_6_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><a NAME="E3:59_1_4_1_1_1_1_2_6_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, , <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_1_1_1_2_6_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green">E172</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1_1_1_1_2_7_1"/>
( ( <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_7_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2_7_1_1"/><i><font color="Green">E173</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_1_1_1_2_7_1_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:59_1_4_1_1_1_1_2_7_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T21" target="_self">Th21</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_1_1_1_2_7_1_1"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T198">XCMPLX_1:198</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_7_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2_7_1_2"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_1_1_1_2_7_1_2"/><b>then </b><i><font color="Green">E184</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_7_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_1_1_1_2_7_1_2_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:59_1_4_1_1_1_1_2_7_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:59_1_4_1_1_1_1_2_7_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_1_1_1_2_7_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_2.html#T21" target="_self">Th21</a>, <a class="ref" href="jgraph_2.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div><a NAME="E4:59_1_4_1_1_1_1_2_7_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E5:59_1_4_1_1_1_1_2_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E6:59_1_4_1_1_1_1_2_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T199">XCMPLX_1:199</a></i>;<br/><b>hence </b><a NAME="E7:59_1_4_1_1_1_1_2_7_1_2"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T192">XCMPLX_1:192</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green">E186</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_8"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1_1_1_1_2_8_1"/>
( ( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_8_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2_8_1_1"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_1_1_1_2_8_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:59_1_4_1_1_1_1_2_8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, , <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_1_1_1_2_8_1_1"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T198">XCMPLX_1:198</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_8_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2_8_1_2"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_1_1_1_2_8_1_2"/><b>then </b><i><font color="Green">E189</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_8_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_1_1_1_2_8_1_2_1"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:59_1_4_1_1_1_1_2_8_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:59_1_4_1_1_1_1_2_8_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:59_1_4_1_1_1_1_2_8_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E4:59_1_4_1_1_1_1_2_8_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E5:59_1_4_1_1_1_1_2_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T22" target="_self">Th22</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E6:59_1_4_1_1_1_1_2_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T199">XCMPLX_1:199</a></i>;<br/><b>hence </b><a NAME="E7:59_1_4_1_1_1_1_2_8_1_2"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T192">XCMPLX_1:192</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1_1_1_1_2_9_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_9_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2_9_1_1"/><i><font color="Green">E191</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_1_1_1_2_9_1_1"/><b>then </b>
( ( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) )
 <b>by </b><i>, , <a class="ref" href="real_2.html#T134">REAL_2:134</a></i>;<br/><a NAME="E3:59_1_4_1_1_1_1_2_9_1_1"/><b>then </b><i><font color="Green">E255</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_9_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_1_1_1_2_9_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:59_1_4_1_1_1_1_2_9_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:59_1_4_1_1_1_1_2_9_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T22" target="_self">Th22</a>, <a class="ref" href="real_2.html#T139">REAL_2:139</a></i>;<br/></div><b>end;</b></div><a NAME="E5:59_1_4_1_1_1_1_2_9_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> = 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 

;<br/><a NAME="E7:59_1_4_1_1_1_1_2_9_1_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><b>hence </b><a NAME="E8:59_1_4_1_1_1_1_2_9_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_9_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_1_1_1_2_9_1_2"/><i><font color="Green">E256</font></i>: 
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_1_1_1_2_9_1_2"/><b>then </b>
( ( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) or ( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T145">XREAL_1:145</a></i>;<br/><a NAME="E3:59_1_4_1_1_1_1_2_9_1_2"/><b>then </b><i><font color="Green">E257</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T128">REAL_2:128</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_1_1_1_2_9_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_1_1_1_2_9_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><a NAME="E2:59_1_4_1_1_1_1_2_9_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T21" target="_self">Th21</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:59_1_4_1_1_1_1_2_9_1_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T22" target="_self">Th22</a>, <a class="ref" href="real_2.html#T139">REAL_2:139</a></i>;<br/></div><b>end;</b></div><b>then </b><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>

;<br/><a NAME="E6:59_1_4_1_1_1_1_2_9_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E7:59_1_4_1_1_1_1_2_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T198">XCMPLX_1:198</a></i>;<br/><a NAME="E8:59_1_4_1_1_1_1_2_9_1_2"/><b>then </b>
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E9:59_1_4_1_1_1_1_2_9_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><b>hence </b><a NAME="E10:59_1_4_1_1_1_1_2_9_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E24:59_1_4_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:59_1_4_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p7</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_1_4_1_2"/><i><font color="Green">E258</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p7</font> &amp; ( ( <font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Maroon">p7</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 ;<br/><a NAME="E4:59_1_4_1_2"/><i><font color="Green">E259</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_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="E1:59_1_4_1_2_1_1"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  or ( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  &amp; not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  ) )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p8</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_1_4_1_2_1_1_1"/><i><font color="Green">E260</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p8</font> &amp; ( ( <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Maroon">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 ;<br/><a NAME="E4:59_1_4_1_2_1_1_1"/><i><font color="Green">E261</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, </i>;<br/><b>set </b><font color="Maroon">qq</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E5:59_1_4_1_2_1_1_1"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E6:59_1_4_1_2_1_1_1"/><b>then </b><i><font color="Green">E262</font></i>: 
( 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><i><font color="Green">E263</font></i>: <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K2">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 

;<br/><i><font color="Green">E264</font></i>: <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K2">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 

;<br/><div><i><font color="Green">E265</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_2_1_1_1_3"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:59_1_4_1_2_1_1_1_3"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:59_1_4_1_2_1_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/></div><b>end;</b></div><a NAME="E10:59_1_4_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T53">XCMPLX_1:53</a></i>;<br/><a NAME="E11:59_1_4_1_2_1_1_1"/><b>then </b><i><font color="Green">E267</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xcmplx_1.html#T53">XCMPLX_1:53</a></i>;<br/><a NAME="E12:59_1_4_1_2_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E13:59_1_4_1_2_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2"/><i><font color="Green">E268</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  &amp; not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p8</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_1_4_1_2_1_1_2"/><i><font color="Green">E269</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p8</font> &amp; ( ( <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 ;<br/><a NAME="E4:59_1_4_1_2_1_1_2"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, </i>;<br/><a NAME="E5:59_1_4_1_2_1_1_2"/><b>then </b><i><font color="Green">E270</font></i>: 
( 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i>, , <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/><div><i><font color="Green">E271</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_2_1_1_2_1"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:59_1_4_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E3:59_1_4_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E273</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_2_1_1_2_2"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:59_1_4_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:59_1_4_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/></div><b>end;</b></div><i><font color="Green">E275</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/><a NAME="E10:59_1_4_1_2_1_1_2"/><b>then </b><i><font color="Green">E276</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xcmplx_1.html#T113">XCMPLX_1:113</a></i>;<br/><a NAME="E11:59_1_4_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E12:59_1_4_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E13:59_1_4_1_2_1_1_2"/><b>then </b><i><font color="Green">E277</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E14:59_1_4_1_2_1_1_2"/><i><font color="Green">E278</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E15:59_1_4_1_2_1_1_2"/><b>then </b><i><font color="Green">E279</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><b>consider </b><font color="Maroon">p9</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E17:59_1_4_1_2_1_1_2"/><i><font color="Green">E280</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p9</font> &amp; ( ( <font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p9</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p9</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p9</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p9</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p9</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p9</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a></i>;<br/><div><i><font color="Green">E281</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1_2_1_1_2_5_1"/>
( ( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_5_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2_5_1_1"/><i><font color="Green">E282</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_2_1_1_2_5_1_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:59_1_4_1_2_1_1_2_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, , <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_2_1_1_2_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_5_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2_5_1_2"/><i><font color="Green">E294</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_5_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_2_1_1_2_5_1_2_1"/><i><font color="Green">E295</font></i>: 
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:59_1_4_1_2_1_1_2_5_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:59_1_4_1_2_1_1_2_5_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:59_1_4_1_2_1_1_2_5_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E3:59_1_4_1_2_1_1_2_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, , <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_2_1_1_2_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green">E296</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1_2_1_1_2_6_1"/>
( ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_6_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2_6_1_1"/><i><font color="Green">E297</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_2_1_1_2_6_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:59_1_4_1_2_1_1_2_6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, , <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_2_1_1_2_6_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_6_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2_6_1_2"/><i><font color="Green">E298</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_6_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_2_1_1_2_6_1_2_1"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:59_1_4_1_2_1_1_2_6_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:59_1_4_1_2_1_1_2_6_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:59_1_4_1_2_1_1_2_6_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E3:59_1_4_1_2_1_1_2_6_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, , <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_2_1_1_2_6_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green">E300</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1_2_1_1_2_7_1"/>
( ( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_7_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2_7_1_1"/><i><font color="Green">E301</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_2_1_1_2_7_1_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:59_1_4_1_2_1_1_2_7_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, , <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_2_1_1_2_7_1_1"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T198">XCMPLX_1:198</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_7_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2_7_1_2"/><i><font color="Green">E302</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_2_1_1_2_7_1_2"/><b>then </b><i><font color="Green">E303</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_7_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_2_1_1_2_7_1_2_1"/><i><font color="Green">E304</font></i>: 
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:59_1_4_1_2_1_1_2_7_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:59_1_4_1_2_1_1_2_7_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:59_1_4_1_2_1_1_2_7_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><a NAME="E4:59_1_4_1_2_1_1_2_7_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E5:59_1_4_1_2_1_1_2_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E6:59_1_4_1_2_1_1_2_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T199">XCMPLX_1:199</a></i>;<br/><b>hence </b><a NAME="E7:59_1_4_1_2_1_1_2_7_1_2"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T192">XCMPLX_1:192</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green">E305</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_8"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1_2_1_1_2_8_1"/>
( ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_8_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2_8_1_1"/><i><font color="Green">E306</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_2_1_1_2_8_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:59_1_4_1_2_1_1_2_8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, , <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E4:59_1_4_1_2_1_1_2_8_1_1"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T198">XCMPLX_1:198</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_8_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2_8_1_2"/><i><font color="Green">E307</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_2_1_1_2_8_1_2"/><b>then </b><i><font color="Green">E308</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_8_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_2_1_1_2_8_1_2_1"/><i><font color="Green">E309</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:59_1_4_1_2_1_1_2_8_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:59_1_4_1_2_1_1_2_8_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:59_1_4_1_2_1_1_2_8_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E4:59_1_4_1_2_1_1_2_8_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E5:59_1_4_1_2_1_1_2_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T21" target="_self">Th21</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E6:59_1_4_1_2_1_1_2_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T199">XCMPLX_1:199</a></i>;<br/><b>hence </b><a NAME="E7:59_1_4_1_2_1_1_2_8_1_2"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T192">XCMPLX_1:192</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_4_1_2_1_1_2_9_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_9_1_1"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2_9_1_1"/><i><font color="Green">E310</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_2_1_1_2_9_1_1"/><b>then </b>
( ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, , <a class="ref" href="real_2.html#T134">REAL_2:134</a></i>;<br/><a NAME="E3:59_1_4_1_2_1_1_2_9_1_1"/><b>then </b><i><font color="Green">E311</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_9_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_2_1_1_2_9_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:59_1_4_1_2_1_1_2_9_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:59_1_4_1_2_1_1_2_9_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, , , <a class="ref" href="jgraph_2.html#T21" target="_self">Th21</a>, <a class="ref" href="real_2.html#T139">REAL_2:139</a></i>;<br/></div><b>end;</b></div><a NAME="E5:59_1_4_1_2_1_1_2_9_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> = 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 

;<br/><a NAME="E7:59_1_4_1_2_1_1_2_9_1_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><b>hence </b><a NAME="E8:59_1_4_1_2_1_1_2_9_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_9_1_2"><b>case </b></a><a NAME="E1:59_1_4_1_2_1_1_2_9_1_2"/><i><font color="Green">E312</font></i>: 
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_1_4_1_2_1_1_2_9_1_2"/><b>then </b>
( ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) or ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T145">XREAL_1:145</a></i>;<br/><a NAME="E3:59_1_4_1_2_1_1_2_9_1_2"/><b>then </b><i><font color="Green">E313</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T128">REAL_2:128</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_4_1_2_1_1_2_9_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_4_1_2_1_1_2_9_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><a NAME="E2:59_1_4_1_2_1_1_2_9_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:59_1_4_1_2_1_1_2_9_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, , , <a class="ref" href="jgraph_2.html#T21" target="_self">Th21</a>, <a class="ref" href="real_2.html#T139">REAL_2:139</a></i>;<br/></div><b>end;</b></div><b>then </b><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span>

;<br/><a NAME="E6:59_1_4_1_2_1_1_2_9_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E7:59_1_4_1_2_1_1_2_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, , <a class="ref" href="xcmplx_1.html#T198">XCMPLX_1:198</a></i>;<br/><a NAME="E8:59_1_4_1_2_1_1_2_9_1_2"/><b>then </b>
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E9:59_1_4_1_2_1_1_2_9_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><b>hence </b><a NAME="E10:59_1_4_1_2_1_1_2_9_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E23:59_1_4_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:59_1_4_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E15:59_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:59"/><b>then </b><i><font color="Green">E314</font></i>: 
 <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E10:59"/><i><font color="Green">E315</font></i>: 
for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> &amp; <font color="Olive">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 holds <br/> not <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Kb</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:59_2"/><i><font color="Green">E316</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 ;<br/>

<b>then consider </b><font color="Maroon">p3</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_2"/><i><font color="Green">E317</font></i>: 
( <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_2_1"/><i><font color="Green">E318</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Kb</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_1_1"/>
( <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> or <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Kb</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_1"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p4</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_2_1_1_1_1"/><i><font color="Green">E320</font></i>: 
( <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_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="E1:59_2_1_1_1_1_1_1"/>
( ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) or ( not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_1_1_1_1"/><i><font color="Green">E321</font></i>: 
( ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_1_1_1_1"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:59_2_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E322</font></i>: 
( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_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="E1:59_2_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E323</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_1_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="E1:59_2_1_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E324</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:59_2_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E325</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:59_2_1_1_1_1_1_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_2_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E326</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/><a NAME="E3:59_2_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E4:59_2_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E5:59_2_1_1_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:59_2_1_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_1_1_1_2"/><i><font color="Green">E327</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_1_1_1_2"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:59_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E328</font></i>: 
( <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E4:59_2_1_1_1_1_1_1_2"/><i><font color="Green">E329</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> implies  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E5:59_2_1_1_1_1_1_1_2"/><i><font color="Green">E330</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> implies  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_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="E1:59_2_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E331</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_2_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="E1:59_2_1_1_1_1_1_1_2_1_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E332</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:59_2_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_1_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_2_1_1_1_1_1_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_2_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E333</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/><a NAME="E3:59_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E4:59_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E5:59_2_1_1_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:59_2_1_1_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:59_2_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Kb</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p4</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_2_1_1_1_2"/><i><font color="Green">E334</font></i>: 
( <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> &amp; ( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_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="E1:59_2_1_1_1_2_1_1"/>
( ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) or ( not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1"/><i><font color="Green">E335</font></i>: 
( ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E336</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:59_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E337</font></i>: 
( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_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="E1:59_2_1_1_1_2_1_1_1_1_1"/>
( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E3:59_2_1_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E338</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_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="E1:59_2_1_1_1_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>hence </b><a NAME="E3:59_2_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E339</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:59_2_1_1_1_2_1_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:59_2_1_1_1_2_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_2"/>
( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_1_1_1_2"/><b>then </b>
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E3:59_2_1_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E340</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_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="E1:59_2_1_1_1_2_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_2_1_1_1_2_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E341</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:59_2_1_1_1_2_1_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:59_2_1_1_1_2_1_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_3"/><i><font color="Green">E342</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">K01</font> = <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <b>by </b><i>, </i>;<br/><a NAME="E3:59_2_1_1_1_2_1_1_1_1_1_3"/><i><font color="Green">E343</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K01</font></span>)</span>
 <b>by </b><i/>;<br/><i><font color="Green">E344</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K01</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K01</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K01</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K01</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K01</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E5:59_2_1_1_1_2_1_1_1_1_1_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K01</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:59_2_1_1_1_2_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E7:59_2_1_1_1_2_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E345</font></i>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K01</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a></i>;<br/><i><font color="Green">E346</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K01</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K01</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K01</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E9:59_2_1_1_1_2_1_1_1_1_1_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K01</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:59_2_1_1_1_2_1_1_1_1_1_3"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>then consider </b><font color="Maroon">p5</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E12:59_2_1_1_1_2_1_1_1_1_1_3"/><i><font color="Green">E348</font></i>: 
( <font color="Maroon">p5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> &amp; ( ( <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p5</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_3_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_3_3_1"/>
( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  )
 </span><b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_3_3_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_3_3_1_1"/>
<font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_1_1_1_3_3_1_1"/><b>then </b><i><font color="Green">E349</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 

;<br/><b>hence </b><a NAME="E4:59_2_1_1_1_2_1_1_1_1_1_3_3_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_3_3_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_3_3_1_2"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_1_1_1_3_3_1_2"/><b>then </b><i><font color="Green">E350</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 

;<br/><b>hence </b><a NAME="E4:59_2_1_1_1_2_1_1_1_1_1_3_3_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:59_2_1_1_1_2_1_1_1_1_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_4"/><i><font color="Green">E351</font></i>: 
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">K01</font> = <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <b>by </b><i>, </i>;<br/><a NAME="E3:59_2_1_1_1_2_1_1_1_1_1_4"/><i><font color="Green">E352</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K01</font></span>)</span>
 <b>by </b><i/>;<br/> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K01</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K01</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K01</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K01</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K01</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E5:59_2_1_1_1_2_1_1_1_1_1_4"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:59_2_1_1_1_2_1_1_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E7:59_2_1_1_1_2_1_1_1_1_1_4"/><b>then </b><i><font color="Green">E353</font></i>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K01</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a></i>;<br/><i><font color="Green">E354</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K01</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K01</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K01</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E9:59_2_1_1_1_2_1_1_1_1_1_4"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K01</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:59_2_1_1_1_2_1_1_1_1_1_4"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K01</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>then consider </b><font color="Maroon">p5</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E12:59_2_1_1_1_2_1_1_1_1_1_4"/><i><font color="Green">E355</font></i>: 
( <font color="Maroon">p5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> &amp; ( ( <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) &amp; <font color="Maroon">p5</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_4_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_4_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_4_3_1"/>
( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  )
 </span><b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="real_2.html#T109">REAL_2:109</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_4_3_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_4_3_1_1"/>
<font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_1_1_1_4_3_1_1"/><b>then </b><i><font color="Green">E356</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 

;<br/><b>hence </b><a NAME="E4:59_2_1_1_1_2_1_1_1_1_1_4_3_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_1_1_1_4_3_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_1_1_1_4_3_1_2"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_1_1_1_4_3_1_2"/><b>then </b><i><font color="Green">E357</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 

;<br/><b>hence </b><a NAME="E4:59_2_1_1_1_2_1_1_1_1_1_4_3_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:59_2_1_1_1_2_1_1_1_1_1_4"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:59_2_1_1_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2"/><i><font color="Green">E358</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E359</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:59_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E360</font></i>: 
( <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E4:59_2_1_1_1_2_1_1_2"/><i><font color="Green">E361</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> implies  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E5:59_2_1_1_1_2_1_1_2"/><i><font color="Green">E362</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> implies  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_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="E1:59_2_1_1_1_2_1_1_2_1_1"/>
( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E3:59_2_1_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E363</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_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="E1:59_2_1_1_1_2_1_1_2_1_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_1_1_1_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>hence </b><a NAME="E3:59_2_1_1_1_2_1_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:59_2_1_1_1_2_1_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E3:59_2_1_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E364</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_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="E1:59_2_1_1_1_2_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:59_2_1_1_1_2_1_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_3"/><i><font color="Green">E365</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_3"/><i><font color="Green">E366</font></i>: 
( ( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">K11</font> = <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <b>by </b><i>, <a class="ref" href="jgraph_2.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:59_2_1_1_1_2_1_1_2_1_1_3"/><i><font color="Green">E367</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T2" target="_self">Th2</a></i>;<br/><i><font color="Green">E368</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K11</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K11</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K11</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K11</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K11</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E6:59_2_1_1_1_2_1_1_2_1_1_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K11</font>
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E7:59_2_1_1_1_2_1_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E8:59_2_1_1_1_2_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E369</font></i>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a></i>;<br/><i><font color="Green">E370</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K11</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K11</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K11</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E10:59_2_1_1_1_2_1_1_2_1_1_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K11</font>
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E11:59_2_1_1_1_2_1_1_2_1_1_3"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>then consider </b><font color="Maroon">p5</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E13:59_2_1_1_1_2_1_1_2_1_1_3"/><i><font color="Green">E371</font></i>: 
( <font color="Maroon">p5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> &amp; ( ( <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Maroon">p5</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_3_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_3_3_1"/>
( <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  )
 </span><b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_3_3_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_3_3_1_1"/>
<font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_3_3_1_1"/><b>then </b><i><font color="Green">E372</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 

;<br/><b>hence </b><a NAME="E4:59_2_1_1_1_2_1_1_2_1_1_3_3_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_3_3_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_3_3_1_2"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_3_3_1_2"/><b>then </b><i><font color="Green">E373</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 

;<br/><b>hence </b><a NAME="E4:59_2_1_1_1_2_1_1_2_1_1_3_3_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:59_2_1_1_1_2_1_1_2_1_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_4"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_4"/><i><font color="Green">E374</font></i>: 
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_4"/><i><font color="Green">E375</font></i>: 
( ( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">K11</font> = <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <b>by </b><i>, <a class="ref" href="jgraph_2.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:59_2_1_1_1_2_1_1_2_1_1_4"/><i><font color="Green">E376</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T2" target="_self">Th2</a></i>;<br/><i><font color="Green">E377</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K11</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K11</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K11</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K11</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K11</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E6:59_2_1_1_1_2_1_1_2_1_1_4"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K11</font>
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E7:59_2_1_1_1_2_1_1_2_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E8:59_2_1_1_1_2_1_1_2_1_1_4"/><b>then </b><i><font color="Green">E378</font></i>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K11</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a></i>;<br/><i><font color="Green">E379</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K11</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K11</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">K11</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E10:59_2_1_1_1_2_1_1_2_1_1_4"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K11</font>
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E11:59_2_1_1_1_2_1_1_2_1_1_4"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>then consider </b><font color="Maroon">p5</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E13:59_2_1_1_1_2_1_1_2_1_1_4"/><i><font color="Green">E380</font></i>: 
( <font color="Maroon">p5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> &amp; ( ( <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p5</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Maroon">p5</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_4_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_4_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_4_3_1"/>
( <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  )
 </span><b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, <a class="ref" href="real_2.html#T109">REAL_2:109</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_4_3_1_1"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_4_3_1_1"/>
<font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_4_3_1_1"/><b>then </b><i><font color="Green">E381</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 

;<br/><b>hence </b><a NAME="E4:59_2_1_1_1_2_1_1_2_1_1_4_3_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1_2_1_1_2_1_1_4_3_1_2"><b>case </b></a><a NAME="E1:59_2_1_1_1_2_1_1_2_1_1_4_3_1_2"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1_2_1_1_2_1_1_4_3_1_2"/><b>then </b><i><font color="Green">E382</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 

;<br/><b>hence </b><a NAME="E4:59_2_1_1_1_2_1_1_2_1_1_4_3_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:59_2_1_1_1_2_1_1_2_1_1_4"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:59_2_1_1_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:59_2_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:59_2"/>
not <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Kb</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:59"/><i><font color="Green">E383</font></i>: 
for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st not <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Kb</font> holds <br/><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:59_3"/>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Kb</font>
 ;<br/>

<a NAME="E2:59_3"/><b>then </b><i><font color="Green">E384</font></i>: 
( not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> &amp; not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Kb</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:59_3"/><b>then </b><i><font color="Green">E385</font></i>: 
not <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:59_3"/><b>then </b>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:59_3"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:59_3"/><b>then </b><i><font color="Green">E386</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p4</font> = <a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_3_1_1"/>
( ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) or ( not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1"><b>case </b></a><a NAME="E1:59_3_1_1_1"/><i><font color="Green">E387</font></i>: 
( ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_1"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:59_3_1_1_1"/><b>then </b><i><font color="Green">E388</font></i>: 
( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_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="E1:59_3_1_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_3_1_1_1_1_1_1"/><i><font color="Green">E389</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E3:59_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E390</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/><a NAME="E4:59_3_1_1_1_1_1_1"/><i><font color="Green">E391</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_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="E1:59_3_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E392</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E393</font></i>: 
 <a href="real_1.html#K1">-</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:59_3_1_1_1_1_1_1_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i>, </i>;<br/><a NAME="E4:59_3_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E394</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:59_3_1_1_1_1_1_1_1_1_1"/>
not <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E6:59_3_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E395</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E7:59_3_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E396</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E8:59_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E397</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E9:59_3_1_1_1_1_1_1_1_1_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E10:59_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:59_3_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E398</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E12:59_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="square_1.html#T76">SQUARE_1:76</a></i>;<br/><a NAME="E13:59_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:59_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E15:59_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/><a NAME="E16:59_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E17:59_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E18:59_3_1_1_1_1_1_1_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E399</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E400</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E3:59_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E401</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div><i><font color="Green">E402</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_3_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:59_3_1_1_1_1_1_1_1_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E3:59_3_1_1_1_1_1_1_1_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:59_3_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/></div><b>end;</b></div><a NAME="E5:59_3_1_1_1_1_1_1_1_1_2"/>
not <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E6:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E403</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E7:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E404</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E8:59_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="square_1.html#T76">SQUARE_1:76</a></i>;<br/><a NAME="E9:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E11:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/><a NAME="E12:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E13:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>;<br/><a NAME="E14:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E15:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E405</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E16:59_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="square_1.html#T76">SQUARE_1:76</a></i>;<br/><a NAME="E17:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E18:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E19:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/><a NAME="E20:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E21:59_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E22:59_3_1_1_1_1_1_1_1_1_2"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, , <a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:59_3_1_1_1_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_3_1_1_1_1_1_2"/><i><font color="Green">E406</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_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="E1:59_3_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_3_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E407</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:59_3_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E408</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E409</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T149">REAL_2:149</a></i>;<br/><a NAME="E3:59_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E410</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E4:59_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E411</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_1_1_2_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="E1:59_3_1_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_1_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E412</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i>, </i>;<br/><a NAME="E3:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1 or 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E413</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1 or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E5:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/>
not <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E6:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E414</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E7:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E8:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E9:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E415</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T191">XCMPLX_1:191</a></i>;<br/><a NAME="E10:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E416</font></i>: 
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="real_2.html#T128">REAL_2:128</a></i>;<br/><a NAME="E11:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E12:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E13:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E417</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E14:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, </i>;<br/><a NAME="E15:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E17:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/><a NAME="E18:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E19:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><b>hence </b><a NAME="E20:59_3_1_1_1_1_1_2_1_1_2_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, , , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_1_1_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E418</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E3:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E4:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E419</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/>
not <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E6:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E420</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E7:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E8:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E9:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E421</font></i>: 
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E10:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E422</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E11:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, </i>;<br/><a NAME="E12:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E14:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/><a NAME="E15:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E16:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>;<br/><a NAME="E17:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E18:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E423</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E19:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, </i>;<br/><a NAME="E20:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E21:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E22:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 ;<br/><a NAME="E23:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E24:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><b>hence </b><a NAME="E25:59_3_1_1_1_1_1_2_1_1_2_1_1_2"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, , , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a>, <a class="ref" href="xcmplx_1.html#T191">XCMPLX_1:191</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:59_3_1_1_1_1_1_2_1_1_2"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_3_1_1_1_1_1_2"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:59_3_1_1_1"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2"><b>case </b></a><a NAME="E1:59_3_1_1_2"/><i><font color="Green">E424</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_2"/><b>then </b>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:59_3_1_1_2"/><b>then </b><i><font color="Green">E425</font></i>: 
( <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E4:59_3_1_1_2"/><i><font color="Green">E426</font></i>: 
( ( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_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="E1:59_3_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_3_1_1_2_1_1_1"/><i><font color="Green">E427</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E3:59_3_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E428</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/><a NAME="E4:59_3_1_1_2_1_1_1"/><i><font color="Green">E429</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E5:59_3_1_1_2_1_1_1"/><i><font color="Green">E430</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_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="E1:59_3_1_1_2_1_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_3_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E431</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E432</font></i>: 
 <a href="real_1.html#K1">-</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:59_3_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E4:59_3_1_1_2_1_1_1_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i>, </i>;<br/><a NAME="E5:59_3_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E434</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:59_3_1_1_2_1_1_1_1_1_1"/>
not <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , <a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, </i>;<br/><a NAME="E7:59_3_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E435</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E8:59_3_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E436</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E9:59_3_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E437</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E10:59_3_1_1_2_1_1_1_1_1_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E11:59_3_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E12:59_3_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E438</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E13:59_3_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, <a class="ref" href="square_1.html#T76">SQUARE_1:76</a></i>;<br/><a NAME="E14:59_3_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:59_3_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E16:59_3_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 ;<br/><a NAME="E17:59_3_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E18:59_3_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E19:59_3_1_1_2_1_1_1_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, , , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_3_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E439</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E440</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div><i><font color="Green">E441</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_2_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_3_1_1_2_1_1_1_1_1_2_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:59_3_1_1_2_1_1_1_1_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:59_3_1_1_2_1_1_1_1_1_2_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:59_3_1_1_2_1_1_1_1_1_2_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/></div><b>end;</b></div><a NAME="E4:59_3_1_1_2_1_1_1_1_1_2"/>
not <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E5:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E442</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E443</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E7:59_3_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T76">SQUARE_1:76</a></i>;<br/><a NAME="E8:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E10:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 ;<br/><a NAME="E11:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E12:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>;<br/><a NAME="E13:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E14:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E444</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E15:59_3_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T76">SQUARE_1:76</a></i>;<br/><a NAME="E16:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E17:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E18:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 ;<br/><a NAME="E19:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E20:59_3_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E21:59_3_1_1_2_1_1_1_1_1_2"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:59_3_1_1_2_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_3_1_1_2_1_1_2"/><i><font color="Green">E445</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_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="E1:59_3_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_3_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E446</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:59_3_1_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_3_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E447</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E448</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T149">REAL_2:149</a></i>;<br/><a NAME="E3:59_3_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E449</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E4:59_3_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E450</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_1_1_2_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="E1:59_3_1_1_2_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_1_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E451</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i>, </i>;<br/><a NAME="E3:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1 or 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E452</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1 or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E5:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/>
not <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E6:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E453</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E7:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E8:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E9:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E454</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T191">XCMPLX_1:191</a></i>;<br/><a NAME="E10:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E455</font></i>: 
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="real_2.html#T128">REAL_2:128</a></i>;<br/><a NAME="E11:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E12:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E456</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E14:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E15:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E17:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 ;<br/><a NAME="E18:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E19:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><b>hence </b><a NAME="E20:59_3_1_1_2_1_1_2_1_1_2_1_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1_1_2_1_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E457</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E458</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E3:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E459</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  or  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E5:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/>
not <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E6:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E460</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E7:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E8:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E9:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E461</font></i>: 
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E10:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E462</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E11:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E12:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E14:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 ;<br/><a NAME="E15:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E16:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>;<br/><a NAME="E17:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E18:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E463</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E19:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E20:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_2.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E21:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><a NAME="E22:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 ;<br/><a NAME="E23:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E24:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><b>hence </b><a NAME="E25:59_3_1_1_2_1_1_2_1_1_2_1_1_2"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_2.html#T13" target="_self">Th13</a>, , , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a>, <a class="ref" href="xcmplx_1.html#T191">XCMPLX_1:191</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:59_3_1_1_2_1_1_2_1_1_2"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_3_1_1_2_1_1_2"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:59_3_1_1_2"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:59_3"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:59"/>
for <font color="Olive">s</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Kb</font> holds <br/><a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:59_4"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Kb</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p4</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:59_4"/><i><font color="Green">E464</font></i>: 
( <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; ( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) )
 <b>by </b><i/>;<br/>
<a NAME="E4:59_4"/><i><font color="Green">E465</font></i>: 
not <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/>
<a NAME="E5:59_4"/><i><font color="Green">E466</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="euclid.html#T58">EUCLID:58</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_4_1_1"/>
( ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) or ( not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1_1"><b>case </b></a><a NAME="E1:59_4_1_1_1"/><i><font color="Green">E467</font></i>: 
( ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) or ( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 ;<br/><div class="add"><a NAME="E2:59_4_1_1_1"/><b>then </b><i><font color="Green">E468</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:59_4_1_1_1"/><i><font color="Green">E469</font></i>: 
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1 ) or ( 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> ) )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_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="E1:59_4_1_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 </span><b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_4_1_1_1_1_1_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_4_1_1_1_1_1_1"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_4_1_1_1_1_1_2"/><i><font color="Green">E470</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T194">XCMPLX_1:194</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> 

;<br/><b>hence </b><a NAME="E3:59_4_1_1_1_1_1_2"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:59_4_1_1_1"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1_2"><b>case </b></a><a NAME="E1:59_4_1_1_2"/><i><font color="Green">E471</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> ) )
 ;<br/><div class="add"><a NAME="E2:59_4_1_1_2"/><b>then </b><i><font color="Green">E472</font></i>: 
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_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="E1:59_4_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_4_1_1_2_1_1_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_4_1_1_2_1_1_1"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_4_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_4_1_1_2_1_1_2"/><i><font color="Green">E473</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T194">XCMPLX_1:194</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> 

;<br/><b>hence </b><a NAME="E3:59_4_1_1_2_1_1_2"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:59_4_1_1_2"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:59_4"/>
<a href="jgraph_2.html#K1" target="_self">Out_In_Sq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:59"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> st <br/>( <font color="Olive">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; ( for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> &amp; <font color="Olive">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 holds <br/> not <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Kb</font> ) &amp; ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st not <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Kb</font> holds <br/><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> ) &amp; ( for <font color="Olive">s</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Kb</font> holds <br/><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> ) )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T10" target="_self">Th10</a>, , </i>;<br/>


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