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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c3">b</font>;<br/>









<b>set </b><font color="Maroon" title="c6">f</font> =  <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">Q</font>;<br/>
<b>assume </b><a NAME="E1:71"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 ;<br/>

<a NAME="E2:71"/><i><font color="Green" title="E56">A2</font></i>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
;<br/>
<a NAME="E3:71"/>
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<a NAME="E1:71_1"/>
<font color="Maroon" title="c8">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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
;<br/>
<a NAME="E2:71_1"/><b>then </b><i><font color="Green" title="E57">A3</font></i>: 
<font color="Maroon" title="c8">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> <a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</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="E3:71_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/>
<a NAME="E4:71_1"/><b>then </b><i><font color="Green" title="E58">A4</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green" title="E59">A5</font></i>: <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:71"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="borsuk_6.html#D6" target="_self" title="BORSUK_6:def.6">Def6</a>, <a class="ref" href="borsuk_6.html#T55" target="_self" title="BORSUK_6:th.55">Th55</a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:71_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_2"/>
( <font color="Maroon" title="c8">p</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="c8">p</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2_1"><b>suppose </b></a><a NAME="E1:71_1_2_1"/><i><font color="Green" title="E60">A6</font></i>: 
<font color="Maroon" title="c8">p</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"><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> = 
<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">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:71_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="borsuk_6.html#E1:71_1_2_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="borsuk_6.html#D7" target="_self" title="BORSUK_6:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:71"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E1:71_1_2_1"><i><font color="Green" title="E60">A6</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="E3:71_1_2_1"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2_2"><b>suppose </b></a><a NAME="E1:71_1_2_2"/><i><font color="Green" title="E60">A7</font></i>: 
<font color="Maroon" title="c8">p</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="E2:71_1_2_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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="ref" href="borsuk_6.html#T7" target="_self" title="BORSUK_6:th.7">Th7</a></i>;<br/><a NAME="E3:71_1_2_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 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> 
 ;<br/><a NAME="E4:71_1_2_2"/><b>then </b><i><font color="Green" title="E61">A8</font></i>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> = 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:71_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="borsuk_6.html#E1:71_1_2_2"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="borsuk_6.html#D7" target="_self" title="BORSUK_6:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:71"><i><font color="Green" title="E55">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#E2:71"><i><font color="Green" title="E56">A2</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> <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="c8">p</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#E4:71_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="borsuk_6.html#E4:71_1_2_2"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:71"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E1:71_1_2_2"><i><font color="Green" title="E60">A7</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="E6:71_1_2_2"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:71"/>
 <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<a href="borsuk_6.html#K3" title="BORSUK_6:func.3">1RP</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
