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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="funct_1.html#V3">constant</a>  <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:73"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 ;<br/>

<a NAME="E2:73"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 
;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<a href="borsuk_6.html#K4" target="_self">2RP</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E3:73"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K5">I[01]</a>  holds <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<a href="borsuk_6.html#K4" target="_self">2RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:73_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
;<br/>
<a NAME="E2:73_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="borsuk_6.html#K4" target="_self">2RP</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:73_1"/>
0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E4:73_1"/><b>then </b><i><font color="Green">E59</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<a href="borsuk_6.html#K4" target="_self">2RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <a href="borsuk_6.html#K4" target="_self">2RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:73"><i><font color="Green">E56</font></i></a>, <a class="ref" href="borsuk_6.html#D6" target="_self">Def6</a>, <a class="ref" href="borsuk_6.html#T56" target="_self">Th56</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K4" target="_self">2RP</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:73_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1"><b>suppose </b></a><a NAME="E1:73_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:73_1_2_1"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E3:73_1_2_1"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 ;<br/><a NAME="E4:73_1_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<a href="borsuk_6.html#K4" target="_self">2RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:73_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="borsuk_6.html#E1:73_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="borsuk_6.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:73"><i><font color="Green">E56</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:73"><i><font color="Green">E57</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:73_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="borsuk_6.html#E4:73_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:73"><i><font color="Green">E56</font></i></a>, <a class="txt" href="borsuk_6.html#E1:73_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E6:73_1_2_1"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<a href="borsuk_6.html#K4" target="_self">2RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_2"><b>suppose </b></a><a NAME="E1:73_1_2_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<a href="borsuk_6.html#K4" target="_self">2RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:73_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="borsuk_6.html#E1:73_1_2_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="borsuk_6.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:73"><i><font color="Green">E56</font></i></a>, <a class="txt" href="borsuk_6.html#E1:73_1_2_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E3:73_1_2_2"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<a href="borsuk_6.html#K4" target="_self">2RP</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:73"/>
 <a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<a href="borsuk_6.html#K4" target="_self">2RP</a>  <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
