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

<span class="kw">let </span><font color="Maroon" title="c1">q4</font>, <font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">O</font>, <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">uq</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Olive" title="b1">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">uq</font> &amp; <font color="Maroon" title="c1">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> &amp; <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">q4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> holds <br/><font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Olive" title="b1">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">O</font>, <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">uq</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c4">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">uq</font> &amp; <font color="Maroon" title="c1">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> &amp; <font color="Maroon" title="c4">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">q4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c4">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:61"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c4">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">uq</font> or  not <font color="Maroon" title="c1">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> or  not <font color="Maroon" title="c4">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> or  not <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">q4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c4">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">uq</font>
 <span class="kw">and </span><br/><a NAME="E3:61"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
<font color="Maroon" title="c1">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font>
 <span class="kw">and </span><br/><a NAME="E4:61"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
<font color="Maroon" title="c4">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">q4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c4">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E5:61"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">q4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c4">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_1"/>
not <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c4">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:61_1"/><span class="kw">then </span>
not  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">O</font>,<font color="Maroon" title="c6">uq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="metric_1.html#T13" title="METRIC_1:th.13">METRIC_1:13</a></span>;<br/><a NAME="E3:61_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="jgraph_1.html#T45" title="JGRAPH_1:th.45">JGRAPH_1:45</a></span>;<br/><a NAME="E4:61_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></span>;<br/><a NAME="E5:61_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T27" title="TOPRNS_1:th.27">TOPRNS_1:27</a></span>;<br/><a NAME="E6:61_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c1">q4</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="toprns_1.html#T27" title="TOPRNS_1:th.27">TOPRNS_1:27</a></span>;<br/><a NAME="E7:61_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">q4</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></span>;<br/><a NAME="E8:61_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">O</font>,<font color="Maroon" title="c5">u</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="jgraph_1.html#T45" title="JGRAPH_1:th.45">JGRAPH_1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E9:61_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="metric_1.html#T13" title="METRIC_1:th.13">METRIC_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:61"/>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c4">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
