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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">Q</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><b>let </b><font color="Maroon">R</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 <b>and </b><br/><a NAME="E3:74"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 ;<br/>

<a NAME="E4:74"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:74"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="borsuk_6.html#K2" target="_self">RePar</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a> ;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>;<br/>
<a NAME="E6:74"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:74_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
;<br/>
<a NAME="E2:74_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="borsuk_6.html#K5" target="_self">3RP</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:74_1"/>
<font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>then </b><i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span> <a href="partfun1.html#K1">*</a> <a href="borsuk_6.html#K5" target="_self">3RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="borsuk_6.html#T6" target="_self">Th6</a>, <a class="ref" href="borsuk_6.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K5" target="_self">3RP</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:74_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_2"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 or ( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 ) or <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="real_1.html#K6">/</a> 4 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1"><b>suppose </b></a><a NAME="E1:74_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i/>;<br/><a NAME="E3:74_1_2_1"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E4:74_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:74_1_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  ;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:74_1_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="borsuk_6.html#E4:74_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="borsuk_6.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i>, , , <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="borsuk_6.html#E4:74_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="txt" href="borsuk_6.html#E4:74_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E7:74_1_2_1"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2"><b>suppose </b></a><a NAME="E1:74_1_2_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 )
 ;<br/><div class="add"><a NAME="E2:74_1_2_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:74_1_2_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:74_1_2_2"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_6.html#E2:74_1_2_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_6.html#E1:74_1_2_2"><i><font color="Green">E65</font></i></a>, </i>;<br/><a NAME="E7:74_1_2_2"/><i><font color="Green">E91</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T15" target="_self">Th15</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, </i>;<br/><a NAME="E9:74_1_2_2"/>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:74_1_2_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="borsuk_6.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i>, , <a class="txt" href="borsuk_6.html#E1:74_1_2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="borsuk_6.html#T16" target="_self">Th16</a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
<b>by </b><i>, , <a class="ref" href="borsuk_6.html#T17" target="_self">Th17</a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , , <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E12:74_1_2_2"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_3"><b>suppose </b></a><a NAME="E1:74_1_2_3"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="real_1.html#K6">/</a> 4
 ;<br/><div class="add"><a NAME="E2:74_1_2_3"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i/>;<br/><a NAME="E4:74_1_2_3"/>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E5:74_1_2_3"/><b>then </b><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="borsuk_6.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
<b>by </b><i>, , , <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , , <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E8:74_1_2_3"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E7:74"/>
 <a href="borsuk_6.html#K2" target="_self">RePar</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>,<a href="borsuk_6.html#K5" target="_self">3RP</a>  <a href="funct_2.html#R2">=</a> <font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
