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<b>let </b><font color="Maroon" title="c4">S</font>, <font color="Maroon" title="c5">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="c6">s1</font>, <font color="Maroon" title="c7">s2</font>, <font color="Maroon" title="c8">s3</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">S</font>;<br/><b>let </b><font color="Maroon" title="c9">t1</font>, <font color="Maroon" title="c10">t2</font>, <font color="Maroon" title="c11">t3</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font>;<br/><b>let </b><font color="Maroon" title="c12">l1</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c9">t1</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>,<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c10">t2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>;<br/><b>let </b><font color="Maroon" title="c13">l2</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c10">t2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>,<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c8">s3</font>,<font color="Maroon" title="c11">t3</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>;<br/><b>let </b><font color="Maroon" title="c14">p1</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c9">t1</font>,<font color="Maroon" title="c10">t2</font>;<br/><b>let </b><font color="Maroon" title="c15">p2</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c10">t2</font>,<font color="Maroon" title="c11">t3</font>;<br/>























<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green" title="E27">A1</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c9">t1</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>,<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c10">t2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green" title="E28">A2</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c10">t2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>,<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c8">s3</font>,<font color="Maroon" title="c11">t3</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c14">p1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c12">l1</font>
 <b>and </b><br/><a NAME="E4:37"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c15">p2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c13">l2</font>
 ;<br/>

<a NAME="E5:37"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c9">t1</font>,<font color="Maroon" title="c10">t2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:37"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="topalg_4.html#T12" target="_self" title="TOPALG_4:th.12">Th12</a></i>;<br/>
<a NAME="E6:37"/><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c10">t2</font>,<font color="Maroon" title="c11">t3</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_4.html#E2:37"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="topalg_4.html#T12" target="_self" title="TOPALG_4:th.12">Th12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ;<br/><a NAME="E1:37_1"/><i><font color="Green" title="E33">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">l2</font></span>)</span> <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="borsuk_1.html#K20" title="BORSUK_1:func.20">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="E2:37_1"/><i><font color="Green" title="E34">A8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">l1</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="borsuk_1.html#K20" title="BORSUK_1:func.20">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="E3:37_1"/><i><font color="Green" title="E35">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">l2</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="borsuk_1.html#K20" title="BORSUK_1:func.20">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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1"/>
( <font color="Maroon" title="c16">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 or <font color="Maroon" title="c16">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1"/><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ;<br/><div class="add"><a NAME="E2:37_1_1_1"/><b>then </b><i><font color="Green" title="E37">A11</font></i>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T6" title="BORSUK_6:th.6">BORSUK_6:6</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">l1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">l2</font></span>)</span></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">l1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">l2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:37_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">x</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:37"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="topalg_4.html#E2:37"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="topalg_4.html#E1:37_1_1_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c14">p1</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="c16">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_4.html#E3:37"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="topalg_4.html#E2:37_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="topalg_4.html#E2:37_1_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font>
<b>by </b><i><a class="txt" href="topalg_4.html#E5:37"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="topalg_4.html#E6:37"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="topalg_4.html#E1:37_1_1_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2"/><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ;<br/><div class="add"><a NAME="E2:37_1_1_2"/><b>then </b><i><font color="Green" title="E37">A13</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="c16">x</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="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T7" title="BORSUK_6:th.7">BORSUK_6:7</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">l1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">l2</font></span>)</span></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">l1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">l2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:37_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">l2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:37"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="topalg_4.html#E2:37"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="topalg_4.html#E1:37_1_1_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c15">p2</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="c16">x</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="topalg_4.html#E4:37"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="topalg_4.html#E3:37_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="topalg_4.html#E2:37_1_1_2"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font>
<b>by </b><i><a class="txt" href="topalg_4.html#E5:37"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="topalg_4.html#E6:37"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="topalg_4.html#E1:37_1_1_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:37"/>
 <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">l2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">p1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p2</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>
