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<span class="kw">let </span><font color="Maroon" title="c1">X</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x0</font>, <font color="Olive" title="b2">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">P</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">x0</font>,<font color="Olive" title="b2">x1</font>  st <font color="Olive" title="b1">x0</font>,<font color="Olive" title="b2">x1</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> <a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Olive" title="b3">P</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <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>  st <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>  holds <br/> <a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Olive" title="b1">P</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  implies  <a href="topalg_1.html#K6" title="TOPALG_1:func.6">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>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:69"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="topalg_1.html#K6" title="TOPALG_1:func.6">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> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Maroon" title="c4">P</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1">(<span class="default"><a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Maroon" title="c4">P</font></span>)</span>) or  not <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1">(<span class="default"><a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Maroon" title="c4">P</font></span>)</span>) or  not <span class="p1">(<span class="default"><a href="topalg_1.html#K6" title="TOPALG_1:func.6">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> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K6" title="TOPALG_1:func.6">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> <font color="Maroon" title="c7">b</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:69"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Maroon" title="c4">P</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:69"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K6" title="TOPALG_1:func.6">pi_1-iso</a> <font color="Maroon" title="c4">P</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:69"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<span class="p1">(<span class="default"><a href="topalg_1.html#K6" title="TOPALG_1:func.6">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> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K6" title="TOPALG_1:func.6">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> <font color="Maroon" title="c7">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">B</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c3">x1</font><span class="kw"> such that </span><br/><a NAME="E6:69"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">x1</font></span>)</span>,<font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="topalg_1.html#T48" target="_self" title="TOPALG_1:th.48">Th48</a></span>;<br/>
<a NAME="E7:69"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<span class="p1">(<span class="default"><a href="topalg_1.html#K6" title="TOPALG_1:func.6">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> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">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> <font color="Maroon" title="c8">B</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> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="topalg_1.html#D6" target="_self" title="TOPALG_1:def.6">Def6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">A</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c3">x1</font><span class="kw"> such that </span><br/><a NAME="E9:69"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">x1</font></span>)</span>,<font color="Maroon" title="c9">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="topalg_1.html#T48" target="_self" title="TOPALG_1:th.48">Th48</a></span>;<br/>
<a NAME="E10:69"/>
<span class="p1">(<span class="default"><a href="topalg_1.html#K6" title="TOPALG_1:func.6">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> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">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> <font color="Maroon" title="c9">A</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> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E9:69"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="topalg_1.html#D6" target="_self" title="TOPALG_1:def.6">Def6</a></span>;<br/>
<a NAME="E11:69"/><span class="kw">then </span>
<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="c9">A</font></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>,<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="c8">B</font></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> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></span>;<br/>
<a NAME="E12:69"/><span class="kw">then </span>
<font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">B</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="topalg_1.html#T28" target="_self" title="TOPALG_1:th.28">Th28</a></span>;<br/>
<a NAME="E13:69"/><span class="kw">then </span>
<font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c8">B</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="topalg_1.html#T30" target="_self" title="TOPALG_1:th.30">Th30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:69"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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