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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">P</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">a</font>;<br/><b>let </b><font color="Maroon" title="c5">Q</font> be  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>;<br/>









<b>assume </b><a NAME="E1:112"/><i><font color="Green" title="E85">A1</font></i>: 
<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">a</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 ;<br/>

<a NAME="E2:112"/><i><font color="Green" title="E86">A2</font></i>: 
<font color="Maroon" title="c4">P</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">S</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c7">S1</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> ;<br/>
<b>set </b><font color="Maroon" title="c8">S2</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> ;<br/>
<b>set </b><font color="Maroon" title="c9">S3</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">e1</font> =  <a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T39" title="YELLOW12:th.39">YELLOW12:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">e2</font> =  <a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T40" title="YELLOW12:th.40">YELLOW12:40</a></i>;<br/>
<a NAME="E5:112"/><i><font color="Green" title="E87">A3</font></i>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">PP</font> = <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font> <b>by </b><i><a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">ff</font> = <font color="Maroon" title="c12">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">e1</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">e1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/>
<a NAME="E8:112"/>
<font color="Maroon" title="c12">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">e1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c14">f</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/>
<a NAME="E10:112"/><i><font color="Green" title="E88">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span> holds  <font color="Maroon" title="c15">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span>;<br/>

<a NAME="E1:112_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span>
 
;<br/>
<a NAME="E2:112_1"/><b>then </b><i><font color="Green" title="E89">A5</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E3:112_1"/><b>then </b><i><font color="Green" title="E90">A6</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
;<br/>
<a NAME="E4:112_1"/><b>then </b><i><font color="Green" title="E91">A7</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E5:112_1"/><b>then </b><i><font color="Green" title="E92">A8</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E6:112_1"/><b>then </b><i><font color="Green" title="E93">A9</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp; <font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:112_1"><i><font color="Green" title="E91">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:112_1"/><b>then </b><i><font color="Green" title="E94">A10</font></i>: 
<font color="Maroon" title="c10">e1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<b>by </b><i><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></i>;<br/>
<a NAME="E8:112_1"/><i><font color="Green" title="E95">A11</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">e1</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:112_1"><i><font color="Green" title="E90">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:112_1"/><i><font color="Green" title="E96">A12</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:112_1"><i><font color="Green" title="E89">A5</font></i></a>, <a class="ref" href="borsuk_6.html#T67" target="_self" title="BORSUK_6:th.67">Th67</a></i>;<br/>
<font color="Maroon" title="c15">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">e1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:112_1"><i><font color="Green" title="E89">A5</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">e1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:112_1"><i><font color="Green" title="E95">A11</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E6:112_1"><i><font color="Green" title="E93">A9</font></i></a>, <a class="txt" href="borsuk_6.html#E7:112_1"><i><font color="Green" title="E94">A10</font></i></a>, <a class="txt" href="borsuk_6.html#E9:112_1"><i><font color="Green" title="E96">A12</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a>, <a class="txt" href="borsuk_6.html#E5:112_1"><i><font color="Green" title="E92">A8</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:112_1"/>
<font color="Maroon" title="c15">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c16">gg</font> = <font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">e2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">gg</font> = <font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">e2</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="borsuk_6.html#E2:112"><i><font color="Green" title="E86">A2</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">g</font> = <font color="Maroon" title="c17">gg</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">g</font> = <font color="Maroon" title="c18">g</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/>
<a NAME="E14:112"/><i><font color="Green" title="E89">A13</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span> holds  <font color="Maroon" title="c19">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>;<br/>

<a NAME="E1:112_2"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>
 
;<br/>
<a NAME="E2:112_2"/><b>then </b><i><font color="Green" title="E90">A14</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E3:112_2"/><b>then </b><i><font color="Green" title="E91">A15</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
;<br/>
<a NAME="E4:112_2"/><b>then </b><i><font color="Green" title="E92">A16</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E5:112_2"/><b>then </b><i><font color="Green" title="E93">A17</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E6:112_2"/><b>then </b><i><font color="Green" title="E94">A18</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp; <font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:112_2"><i><font color="Green" title="E92">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:112_2"/><b>then </b><i><font color="Green" title="E95">A19</font></i>: 
<font color="Maroon" title="c11">e2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<b>by </b><i><a class="ref" href="funct_3.html#D6" title="FUNCT_3:def.6">FUNCT_3:def 6</a></i>;<br/>
<a NAME="E8:112_2"/><i><font color="Green" title="E96">A20</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:112_2"><i><font color="Green" title="E94">A18</font></i></a>, <a class="ref" href="jordan5b.html#T4" title="JORDAN5B:th.4">JORDAN5B:4</a></i>;<br/>
<a NAME="E9:112_2"/><i><font color="Green" title="E97">A21</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">e2</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:112_2"><i><font color="Green" title="E91">A15</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon" title="c19">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font> = 
<font color="Maroon" title="c17">gg</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:112_2"><i><font color="Green" title="E90">A14</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="txt" href="borsuk_6.html#E5:112_2"><i><font color="Green" title="E93">A17</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:112_2"><i><font color="Green" title="E93">A17</font></i></a>, <a class="txt" href="borsuk_6.html#E7:112_2"><i><font color="Green" title="E95">A19</font></i></a>, <a class="txt" href="borsuk_6.html#E9:112_2"><i><font color="Green" title="E97">A21</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E8:112_2"><i><font color="Green" title="E96">A20</font></i></a>, <a class="ref" href="borsuk_2.html#D6" title="BORSUK_2:def.6">BORSUK_2:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E11:112_2"/>
<font color="Maroon" title="c19">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c20">hh</font> = <font color="Maroon" title="c12">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">e1</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">h</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">e1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/>
<a NAME="E16:112"/>
<font color="Maroon" title="c12">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">e1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">h</font> = <font color="Maroon" title="c21">h</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/>
<a NAME="E18:112"/><i><font color="Green" title="E90">A22</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span> holds  <font color="Maroon" title="c22">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>;<br/>

<a NAME="E1:112_3"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>
 
;<br/>
<a NAME="E2:112_3"/><b>then </b><i><font color="Green" title="E91">A23</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E3:112_3"/><b>then </b><i><font color="Green" title="E92">A24</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
;<br/>
<a NAME="E4:112_3"/><b>then </b><i><font color="Green" title="E93">A25</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E5:112_3"/><b>then </b><i><font color="Green" title="E94">A26</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E6:112_3"/><b>then </b><i><font color="Green" title="E95">A27</font></i>: 
( <font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp; <font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:112_3"><i><font color="Green" title="E93">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:112_3"/><b>then </b><i><font color="Green" title="E96">A28</font></i>: 
<font color="Maroon" title="c10">e1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<b>by </b><i><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></i>;<br/>
<a NAME="E8:112_3"/><i><font color="Green" title="E97">A29</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">e1</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:112_3"><i><font color="Green" title="E92">A24</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:112_3"/><i><font color="Green" title="E98">A30</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:112_3"><i><font color="Green" title="E91">A23</font></i></a>, <a class="ref" href="borsuk_6.html#T68" target="_self" title="BORSUK_6:th.68">Th68</a></i>;<br/>
<font color="Maroon" title="c22">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">e1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:112_3"><i><font color="Green" title="E91">A23</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">e1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:112_3"><i><font color="Green" title="E97">A29</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="txt" href="borsuk_6.html#E5:112_3"><i><font color="Green" title="E94">A26</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E6:112_3"><i><font color="Green" title="E95">A27</font></i></a>, <a class="txt" href="borsuk_6.html#E7:112_3"><i><font color="Green" title="E96">A28</font></i></a>, <a class="txt" href="borsuk_6.html#E9:112_3"><i><font color="Green" title="E98">A30</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E11:112_3"/>
<font color="Maroon" title="c22">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c23">S12</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">S12</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> ;<br/>
<a NAME="E20:112"/><i><font color="Green" title="E91">A31</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c24">S12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E21:112"/><i><font color="Green" title="E92">A32</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E22:112"/><i><font color="Green" title="E93">A33</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">s1</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span>, <font color="Maroon" title="c26">s2</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c24">S12</font> <b>by </b><i><a class="txt" href="borsuk_6.html#E20:112"><i><font color="Green" title="E91">A31</font></i></a>, <a class="txt" href="borsuk_6.html#E21:112"><i><font color="Green" title="E92">A32</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E24:112"/><i><font color="Green" title="E94">A34</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">S12</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E20:112"><i><font color="Green" title="E91">A31</font></i></a>, <a class="txt" href="borsuk_6.html#E22:112"><i><font color="Green" title="E93">A33</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E25:112"/><i><font color="Green" title="E95">A35</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span></span>)</span> holds <br/><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:112_4"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:112_4"/><b>then </b>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E3:112_4"/><b>then </b><i><font color="Green" title="E96">A36</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c28">r</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E5:112_4"/><i><font color="Green" title="E97">A37</font></i>: 
( <font color="Maroon" title="c28">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p</font> &amp; <font color="Maroon" title="c28">r</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">r</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="borsuk_6.html#T65" target="_self" title="BORSUK_6:th.65">Th65</a></i>;<br/>
<a NAME="E6:112_4"/><i><font color="Green" title="E98">A38</font></i>: 
( <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:112_4"><i><font color="Green" title="E96">A36</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c29">pp</font> = <font color="Maroon" title="c27">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E8:112_4"/><i><font color="Green" title="E99">A39</font></i>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">r</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">r</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:112_4"><i><font color="Green" title="E97">A37</font></i></a></i>;<br/>
<a NAME="E9:112_4"/><i><font color="Green" title="E100">A40</font></i>: 
<font color="Maroon" title="c29">pp</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:112_4"><i><font color="Green" title="E98">A38</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">p</font> = 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">pp</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E10:112"><i><font color="Green" title="E88">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">p</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E14:112"><i><font color="Green" title="E89">A13</font></i></a>, <a class="txt" href="borsuk_6.html#E5:112_4"><i><font color="Green" title="E97">A37</font></i></a>, <a class="txt" href="borsuk_6.html#E8:112_4"><i><font color="Green" title="E99">A39</font></i></a>, <a class="txt" href="borsuk_6.html#E9:112_4"><i><font color="Green" title="E100">A40</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:112_4"/>
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">p</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E26:112"/>
 <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E27:112"/><b>then </b><i><font color="Green" title="E96">A41</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>  is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c24">S12</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T29" title="TOPMETR:th.29">TOPMETR:29</a></i>;<br/>
<a NAME="E28:112"/>
 <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E29:112"/><b>then </b><i><font color="Green" title="E97">A42</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>  is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c24">S12</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T29" title="TOPMETR:th.29">TOPMETR:29</a></i>;<br/>
<a NAME="E30:112"/><i><font color="Green" title="E98">A43</font></i>: 
<font color="Maroon" title="c25">s1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E21:112"><i><font color="Green" title="E92">A32</font></i></a>, <a class="ref" href="tops_2.html#T34" title="TOPS_2:th.34">TOPS_2:34</a></i>;<br/>
<a NAME="E31:112"/>
<font color="Maroon" title="c26">s2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E22:112"><i><font color="Green" title="E93">A33</font></i></a>, <a class="ref" href="tops_2.html#T34" title="TOPS_2:th.34">TOPS_2:34</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c27">fg</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c24">S12</font>,<font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E33:112"/><i><font color="Green" title="E99">A44</font></i>: 
( <font color="Maroon" title="c27">fg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c19">g</font> &amp; <font color="Maroon" title="c27">fg</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E24:112"><i><font color="Green" title="E94">A34</font></i></a>, <a class="txt" href="borsuk_6.html#E25:112"><i><font color="Green" title="E95">A35</font></i></a>, <a class="txt" href="borsuk_6.html#E27:112"><i><font color="Green" title="E96">A41</font></i></a>, <a class="txt" href="borsuk_6.html#E29:112"><i><font color="Green" title="E97">A42</font></i></a>, <a class="txt" href="borsuk_6.html#E30:112"><i><font color="Green" title="E98">A43</font></i></a>, <a class="ref" href="jgraph_2.html#T9" title="JGRAPH_2:th.9">JGRAPH_2:9</a></i>;<br/>
<a NAME="E34:112"/><i><font color="Green" title="E100">A45</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">s12</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">S12</font>, <font color="Maroon" title="c29">s3</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E36:112"/><i><font color="Green" title="E101">A46</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">S12</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E20:112"><i><font color="Green" title="E91">A31</font></i></a>, <a class="txt" href="borsuk_6.html#E34:112"><i><font color="Green" title="E100">A45</font></i></a>, <a class="ref" href="borsuk_6.html#T64" target="_self" title="BORSUK_6:th.64">Th64</a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E37:112"/><i><font color="Green" title="E102">A47</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">S12</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span></span>)</span> holds <br/><font color="Maroon" title="c27">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:112_5"/>
<font color="Maroon" title="c30">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">S12</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:112_5"/><b>then </b><i><font color="Green" title="E103">A48</font></i>: 
<font color="Maroon" title="c30">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E20:112"><i><font color="Green" title="E91">A31</font></i></a>, <a class="txt" href="borsuk_6.html#E34:112"><i><font color="Green" title="E100">A45</font></i></a>, <a class="ref" href="borsuk_6.html#T80" target="_self" title="BORSUK_6:th.80">Th80</a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E3:112_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T74" target="_self" title="BORSUK_6:th.74">Th74</a>, <a class="ref" href="borsuk_6.html#T75" target="_self" title="BORSUK_6:th.75">Th75</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:112_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E5:112_5"/><b>then </b><i><font color="Green" title="E104">A49</font></i>: 
<font color="Maroon" title="c30">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:112_5"><i><font color="Green" title="E103">A48</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c31">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E7:112_5"/><i><font color="Green" title="E105">A50</font></i>: 
( <font color="Maroon" title="c31">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p</font> &amp; <font color="Maroon" title="c31">q</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">q</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <b>by </b><i><a class="ref" href="borsuk_6.html#T66" target="_self" title="BORSUK_6:th.66">Th66</a></i>;<br/>
<a NAME="E8:112_5"/><i><font color="Green" title="E106">A51</font></i>: 
( <font color="Maroon" title="c30">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>  &amp; <font color="Maroon" title="c30">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:112_5"><i><font color="Green" title="E104">A49</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c32">pp</font> = <font color="Maroon" title="c30">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E10:112_5"/><i><font color="Green" title="E107">A52</font></i>: 
<font color="Maroon" title="c32">pp</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:112_5"><i><font color="Green" title="E106">A51</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E11:112_5"/>
( <font color="Maroon" title="c32">pp</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp; <font color="Maroon" title="c32">pp</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E7:112_5"><i><font color="Green" title="E105">A50</font></i></a>, <a class="ref" href="borsuk_6.html#T31" target="_self" title="BORSUK_6:th.31">Th31</a></i>;<br/>
<a NAME="E12:112_5"/><b>then </b><i><font color="Green" title="E108">A53</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">pp</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="jordan5b.html#T4" title="JORDAN5B:th.4">JORDAN5B:4</a></i>;<br/>
<a NAME="E13:112_5"/>
<font color="Maroon" title="c30">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:112_5"><i><font color="Green" title="E106">A51</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E14:112_5"/><b>then </b>
<font color="Maroon" title="c30">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">g</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><font color="Maroon" title="c27">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">p</font> = 
<font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">p</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:112"><i><font color="Green" title="E99">A44</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">pp</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E14:112"><i><font color="Green" title="E89">A13</font></i></a>, <a class="txt" href="borsuk_6.html#E10:112_5"><i><font color="Green" title="E107">A52</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c32">pp</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E12:112_5"><i><font color="Green" title="E108">A53</font></i></a>, <a class="ref" href="borsuk_2.html#D6" title="BORSUK_2:def.6">BORSUK_2:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">p</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:112"><i><font color="Green" title="E90">A22</font></i></a>, <a class="txt" href="borsuk_6.html#E7:112_5"><i><font color="Green" title="E105">A50</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E16:112_5"/>
<font color="Maroon" title="c27">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">p</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E38:112"/><i><font color="Green" title="E103">A54</font></i>: 
<font color="Maroon" title="c28">s12</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E20:112"><i><font color="Green" title="E91">A31</font></i></a>, <a class="ref" href="tops_1.html#T36" title="TOPS_1:th.36">TOPS_1:36</a></i>;<br/>
<a NAME="E39:112"/>
<font color="Maroon" title="c29">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c30">H</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E41:112"/><i><font color="Green" title="E104">A55</font></i>: 
( <font color="Maroon" title="c30">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">fg</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c22">h</font> &amp; <font color="Maroon" title="c30">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E33:112"><i><font color="Green" title="E99">A44</font></i></a>, <a class="txt" href="borsuk_6.html#E36:112"><i><font color="Green" title="E101">A46</font></i></a>, <a class="txt" href="borsuk_6.html#E37:112"><i><font color="Green" title="E102">A47</font></i></a>, <a class="txt" href="borsuk_6.html#E38:112"><i><font color="Green" title="E103">A54</font></i></a>, <a class="ref" href="jgraph_2.html#T9" title="JGRAPH_2:th.9">JGRAPH_2:9</a></i>;<br/>
<a NAME="E42:112"/><i><font color="Green" title="E105">A56</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c31">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>thus </b><a NAME="E1:112_6"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c31">s</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_6_1"><b>proof </b></a><div class="add">

<a NAME="E1:112_6_1"/><i><font color="Green" title="E106">A57</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">s</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:112_6_1_1"/>
( <font color="Maroon" title="c31">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 or <font color="Maroon" title="c31">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 or <font color="Maroon" title="c31">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_6_1_1_1"><b>suppose </b></a><a NAME="E1:112_6_1_1_1"/><i><font color="Green" title="E107">A58</font></i>: 
<font color="Maroon" title="c31">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:112_6_1_1_1"/><b>then </b><i><font color="Green" title="E108">A59</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T78" target="_self" title="BORSUK_6:th.78">Th78</a></i>;<br/><a NAME="E3:112_6_1_1_1"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:112_6_1_1_1"><i><font color="Green" title="E107">A58</font></i></a>, <a class="ref" href="borsuk_6.html#T79" target="_self" title="BORSUK_6:th.79">Th79</a></i>;<br/><a NAME="E4:112_6_1_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E5:112_6_1_1_1"/><b>then </b><i><font color="Green" title="E109">A60</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">h</font>
 ;<br/><a NAME="E6:112_6_1_1_1"/><i><font color="Green" title="E110">A61</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E2:112_6_1_1_1"><i><font color="Green" title="E108">A59</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E7:112_6_1_1_1"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:112_6_1_1_1"><i><font color="Green" title="E107">A58</font></i></a>, <a class="ref" href="borsuk_6.html#T79" target="_self" title="BORSUK_6:th.79">Th79</a></i>;<br/><a NAME="E8:112_6_1_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E9:112_6_1_1_1"/><b>then </b><i><font color="Green" title="E111">A62</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">g</font>
 ;<br/><font color="Maroon" title="c30">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c27">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E41:112"><i><font color="Green" title="E104">A55</font></i></a>, <a class="txt" href="borsuk_6.html#E5:112_6_1_1_1"><i><font color="Green" title="E109">A60</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:112"><i><font color="Green" title="E99">A44</font></i></a>, <a class="txt" href="borsuk_6.html#E9:112_6_1_1_1"><i><font color="Green" title="E111">A62</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">s</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E10:112"><i><font color="Green" title="E88">A4</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_6_1"><i><font color="Green" title="E106">A57</font></i></a>, <a class="txt" href="borsuk_6.html#E6:112_6_1_1_1"><i><font color="Green" title="E110">A61</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_6_1_1_1"><i><font color="Green" title="E107">A58</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E11:112_6_1_1_1"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c31">s</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_6_1_1_2"><b>suppose </b></a><a NAME="E1:112_6_1_1_2"/><i><font color="Green" title="E107">A63</font></i>: 
<font color="Maroon" title="c31">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:112_6_1_1_2"/><b>then </b><i><font color="Green" title="E108">A64</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T74" target="_self" title="BORSUK_6:th.74">Th74</a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E3:112_6_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c30">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E41:112"><i><font color="Green" title="E104">A55</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:112"><i><font color="Green" title="E90">A22</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_6_1"><i><font color="Green" title="E106">A57</font></i></a>, <a class="txt" href="borsuk_6.html#E2:112_6_1_1_2"><i><font color="Green" title="E108">A64</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">b</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_6_1_1_2"><i><font color="Green" title="E107">A63</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_6_1_1_2"><i><font color="Green" title="E107">A63</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E5:112_6_1_1_2"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c31">s</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_6_1_1_3"><b>suppose </b></a><a NAME="E1:112_6_1_1_3"/><i><font color="Green" title="E107">A65</font></i>: 
<font color="Maroon" title="c31">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:112_6_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T77" target="_self" title="BORSUK_6:th.77">Th77</a></i>;<br/><a NAME="E3:112_6_1_1_3"/><b>then </b><i><font color="Green" title="E108">A66</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E4:112_6_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c30">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E41:112"><i><font color="Green" title="E104">A55</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:112"><i><font color="Green" title="E90">A22</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_6_1"><i><font color="Green" title="E106">A57</font></i></a>, <a class="txt" href="borsuk_6.html#E3:112_6_1_1_3"><i><font color="Green" title="E108">A66</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_6_1_1_3"><i><font color="Green" title="E107">A65</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E6:112_6_1_1_3"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c31">s</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:112_6"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c31">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_6_2"><b>proof </b></a><div class="add">

<a NAME="E1:112_6_2"/><i><font color="Green" title="E106">A67</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">s</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E2:112_6_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:112_6_2"/><b>then </b><i><font color="Green" title="E107">A68</font></i>: 
( 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c31">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font> )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:112_6_2_1"/>
( <font color="Maroon" title="c31">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or <font color="Maroon" title="c31">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_6_2_1_1"><b>suppose </b></a><a NAME="E1:112_6_2_1_1"/>
<font color="Maroon" title="c31">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:112_6_2_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T71" target="_self" title="BORSUK_6:th.71">Th71</a></i>;<br/><a NAME="E3:112_6_2_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E4:112_6_2_1_1"/><b>then </b><i><font color="Green" title="E108">A69</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">h</font>
 ;<br/><a NAME="E5:112_6_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T73" target="_self" title="BORSUK_6:th.73">Th73</a></i>;<br/><a NAME="E6:112_6_2_1_1"/><b>then </b><i><font color="Green" title="E109">A70</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E7:112_6_2_1_1"/><b>then </b><i><font color="Green" title="E110">A71</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon" title="c30">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c27">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E41:112"><i><font color="Green" title="E104">A55</font></i></a>, <a class="txt" href="borsuk_6.html#E4:112_6_2_1_1"><i><font color="Green" title="E108">A69</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:112"><i><font color="Green" title="E99">A44</font></i></a>, <a class="txt" href="borsuk_6.html#E7:112_6_2_1_1"><i><font color="Green" title="E110">A71</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E14:112"><i><font color="Green" title="E89">A13</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_6_2"><i><font color="Green" title="E106">A67</font></i></a>, <a class="txt" href="borsuk_6.html#E6:112_6_2_1_1"><i><font color="Green" title="E109">A70</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:112"><i><font color="Green" title="E87">A3</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:112_6_2"><i><font color="Green" title="E107">A68</font></i></a>, <a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>
;<br/><b>hence </b><a NAME="E9:112_6_2_1_1"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c31">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_6_2_1_2"><b>suppose </b></a><a NAME="E1:112_6_2_1_2"/><i><font color="Green" title="E108">A72</font></i>: 
<font color="Maroon" title="c31">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:112_6_2_1_2"/><b>then </b><i><font color="Green" title="E109">A73</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T74" target="_self" title="BORSUK_6:th.74">Th74</a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E3:112_6_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c30">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E41:112"><i><font color="Green" title="E104">A55</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:112"><i><font color="Green" title="E90">A22</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_6_2"><i><font color="Green" title="E106">A67</font></i></a>, <a class="txt" href="borsuk_6.html#E2:112_6_2_1_2"><i><font color="Green" title="E109">A73</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_6_2_1_2"><i><font color="Green" title="E108">A72</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:112"><i><font color="Green" title="E87">A3</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:112_6_2"><i><font color="Green" title="E107">A68</font></i></a>, <a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>
;<br/><b>hence </b><a NAME="E5:112_6_2_1_2"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c31">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">s</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<a NAME="E43:112"/>
for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c31">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>thus </b><a NAME="E1:112_7"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Maroon" title="c31">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_7_1"><b>proof </b></a><div class="add">

<a NAME="E1:112_7_1"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c32">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c31">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="txt" href="borsuk_6.html#E93"><i><font color="Green" title="E77">Lm1</font></i></a></i>;<br/>
<a NAME="E3:112_7_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T69" target="_self" title="BORSUK_6:th.69">Th69</a></i>;<br/>
<a NAME="E4:112_7_1"/><b>then </b><i><font color="Green" title="E106">A74</font></i>: 
<font color="Maroon" title="c32">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E5:112_7_1"/><i><font color="Green" title="E107">A75</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c32">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">t</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E6:112_7_1"/><b>then </b>
not <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T68" target="_self" title="BORSUK_6:th.68">Th68</a></i>;<br/>
<a NAME="E7:112_7_1"/><b>then </b>
not <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E8:112_7_1"/><b>then </b><i><font color="Green" title="E108">A76</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c31">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">h</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:112_7_1_1"/>
( <font color="Maroon" title="c31">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or <font color="Maroon" title="c31">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_7_1_1_1"><b>suppose </b></a><a NAME="E1:112_7_1_1_1"/>
<font color="Maroon" title="c31">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:112_7_1_1_1"/><b>then </b>
not <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T70" target="_self" title="BORSUK_6:th.70">Th70</a></i>;<br/><a NAME="E3:112_7_1_1_1"/><b>then </b>
not <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E4:112_7_1_1_1"/><b>then </b>
not <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">g</font>
 ;<br/><b>then </b><font color="Maroon" title="c27">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c31">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c31">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:112"><i><font color="Green" title="E99">A44</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E10:112"><i><font color="Green" title="E88">A4</font></i></a>, <a class="txt" href="borsuk_6.html#E4:112_7_1"><i><font color="Green" title="E106">A74</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E5:112_7_1"><i><font color="Green" title="E107">A75</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
;<br/><b>hence </b><a NAME="E6:112_7_1_1_1"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Maroon" title="c31">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E41:112"><i><font color="Green" title="E104">A55</font></i></a>, <a class="txt" href="borsuk_6.html#E8:112_7_1"><i><font color="Green" title="E108">A76</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_7_1_1_2"><b>suppose </b></a><a NAME="E1:112_7_1_1_2"/><i><font color="Green" title="E109">A77</font></i>: 
<font color="Maroon" title="c31">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:112_7_1_1_2"/><b>then </b><i><font color="Green" title="E110">A78</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> </span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T72" target="_self" title="BORSUK_6:th.72">Th72</a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E3:112_7_1_1_2"/><b>then </b>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c27">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c31">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:112"><i><font color="Green" title="E99">A44</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_7_1_1_2"><i><font color="Green" title="E109">A77</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E14:112"><i><font color="Green" title="E89">A13</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_7_1_1_2"><i><font color="Green" title="E109">A77</font></i></a>, <a class="txt" href="borsuk_6.html#E2:112_7_1_1_2"><i><font color="Green" title="E110">A78</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E5:112_7_1"><i><font color="Green" title="E107">A75</font></i></a>, <a class="txt" href="borsuk_6.html#E1:112_7_1_1_2"><i><font color="Green" title="E109">A77</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
;<br/><b>hence </b><a NAME="E5:112_7_1_1_2"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Maroon" title="c31">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E41:112"><i><font color="Green" title="E104">A55</font></i></a>, <a class="txt" href="borsuk_6.html#E8:112_7_1"><i><font color="Green" title="E108">A76</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:112_7"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c31">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_7_2"><b>proof </b></a><div class="add">

<a NAME="E1:112_7_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c32">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c31">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="txt" href="borsuk_6.html#E93"><i><font color="Green" title="E77">Lm1</font></i></a></i>;<br/>
<a NAME="E3:112_7_2"/><i><font color="Green" title="E106">A79</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T76" target="_self" title="BORSUK_6:th.76">Th76</a></i>;<br/>
<a NAME="E4:112_7_2"/><i><font color="Green" title="E107">A80</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c32">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">t</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E5:112_7_2"/><i><font color="Green" title="E108">A81</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:112_7_2"><i><font color="Green" title="E106">A79</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E6:112_7_2"/><b>then </b><i><font color="Green" title="E109">A82</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c31">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">h</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon" title="c22">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c31">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:112"><i><font color="Green" title="E90">A22</font></i></a>, <a class="txt" href="borsuk_6.html#E5:112_7_2"><i><font color="Green" title="E108">A81</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:112"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E4:112_7_2"><i><font color="Green" title="E107">A80</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E8:112_7_2"/>
<font color="Maroon" title="c30">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c31">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E41:112"><i><font color="Green" title="E104">A55</font></i></a>, <a class="txt" href="borsuk_6.html#E6:112_7_2"><i><font color="Green" title="E109">A82</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E44:112"/>
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="borsuk_6.html#E41:112"><i><font color="Green" title="E104">A55</font></i></a>, <a class="txt" href="borsuk_6.html#E42:112"><i><font color="Green" title="E105">A56</font></i></a>, <a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a></i>;<br/>


</div>
