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

<b>let </b><font color="Maroon" title="c1">X</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">x0</font>, <font color="Maroon" title="c3">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</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">x0</font>,<font color="Maroon" title="c3">x1</font>;<br/>







<b>assume </b><a NAME="E1:68"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c2">x0</font>,<font color="Maroon" title="c3">x1</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c5">f</font> =  <a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font>;<br/>
<b>set </b><font color="Maroon" title="c6">g</font> =  <a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</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>;<br/>
<a NAME="E2:68"/><i><font color="Green" title="E44">A2</font></i>: 
 <a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:68"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="topalg_1.html#T52" target="_self" title="TOPALG_1:th.52">Th52</a></i>;<br/>
<a NAME="E3:68"/>
 <a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:68"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="topalg_1.html#T53" target="_self" title="TOPALG_1:th.53">Th53</a></i>;<br/>
<a NAME="E4:68"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</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 <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:68"/><b>then </b><i><font color="Green" title="E45">A3</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:68"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E6:68"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c8">Q</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">x0</font><b> such that </b><br/><a NAME="E2:68_1"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>,<font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self" title="TOPALG_1:th.48">Th48</a></i>;<br/>
<a NAME="E3:68_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</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 <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">x1</font></span>)</span>
 
<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="E4:68_1"/><b>then </b><i><font color="Green" title="E47">A5</font></i>: 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">x1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c8">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <span class="p3">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span></span>)</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> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self" title="TOPALG_1:th.48">Th48</a></i>;<br/>
<a NAME="E5:68_1"/>
 <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</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_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:68"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="borsuk_6.html#T50" title="BORSUK_6:th.50">BORSUK_6:50</a></i>;<br/>
<a NAME="E6:68_1"/><b>then </b><i><font color="Green" title="E48">A6</font></i>: 
<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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="c8">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p3">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <span class="p4">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.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>,<font color="Maroon" title="c8">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:68"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="topalg_1.html#T42" target="_self" title="TOPALG_1:th.42">Th42</a></i>;<br/>
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</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"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p2">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="c8">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p3">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <span class="p4">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">x0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="c8">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p4">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <span class="p5">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <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></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:68"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="topalg_1.html#D4" target="_self" title="TOPALG_1:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="txt" href="topalg_1.html#E2:68_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="topalg_1.html#E6:68_1"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">x1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c8">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <span class="p3">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E2:68"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="topalg_1.html#E5:68"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="topalg_1.html#E4:68_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:68"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="topalg_1.html#E2:68_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="topalg_1.html#D4" target="_self" title="TOPALG_1:def.4">Def4</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:68"/>
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_1.html#K4" title="TOPALG_1:func.4">pi_1-iso</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>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
