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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[ <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[ <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="real_1.html#K6">/</a> 4;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>( <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <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">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>( <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></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>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>( <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1;<br/>
<a NAME="E1:69_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] implies  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">c</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] implies  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>] ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">c</font>] implies  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>] ) )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:69_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K5">I[01]</a>   holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">c</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>] )
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:69_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; ( for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">c</font>) ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">c</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">c</font>) ) &amp; ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">c</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="borsuk_6.html#S1" target="_self">BORSUK_6:sch 1</a>(<a class="ref" href="borsuk_6.html#T10" target="_self">Th10</a>, <a class="ref" href="borsuk_6.html#T12" target="_self">Th12</a>);<br/></i>
<a NAME="E5:69_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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>  holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:69_1_1_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/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 ;<br/><div class="add"><a NAME="E2:69_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E3:69_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:69_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>]
 ;<br/><div class="add"><a NAME="E2:69_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>)
 <b>by </b><i><a class="ref" href="borsuk_6.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E3:69_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:69_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_1_1_1_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>]
 ;<br/><div class="add"><a NAME="E2:69_1_1_1_1_3"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" 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/><a NAME="E3:69_1_1_1_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <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
 <b>by </b><i><a class="ref" href="borsuk_6.html#T13" target="_self">Th13</a>, </i>;<br/><a NAME="E4:69_1_1_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T14" target="_self">Th14</a>, </i>;<br/><b>hence </b><a NAME="E5:69_1_1_1_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_6.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E7:69_1_1"/>
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 <br/> ( ( <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <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="Olive">x</font> ) &amp; ( <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">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> ) &amp; ( <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="real_1.html#K6">/</a> 4 implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 ) )
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E8:69_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a>  st <br/>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 <br/> ( ( <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <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="Olive">x</font> ) &amp; ( <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">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> ) &amp; ( <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="real_1.html#K6">/</a> 4 implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 ) )
 ;<br/>


</div>
