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<span class="kw">let </span><font color="Maroon" title="c1">S</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">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</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#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">s</font></span>)</span><br/> for <font color="Olive" title="b4">P</font>, <font color="Olive" title="b5">Q</font> being  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Olive" title="b1">s</font>  st <font color="Olive" title="b2">x</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">S</font>,<font color="Olive" title="b1">s</font></span>)</span>,<font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b3">y</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">S</font>,<font color="Olive" title="b1">s</font></span>)</span>,<font color="Olive" title="b5">Q</font> holds <br/><font color="Olive" title="b2">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">y</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">S</font>,<font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b5">Q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</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">x</font>, <font color="Olive" title="b2">y</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#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span><br/> for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">s</font>  st <font color="Olive" title="b1">x</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Olive" title="b4">Q</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b4">Q</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">X</font> =  <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>; <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>, <font color="Olive" title="b2">Q</font> being  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">s</font>  st <font color="Maroon" title="c4">x</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c5">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Olive" title="b2">Q</font> holds <br/><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b2">Q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c7">Q</font> be   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">s</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="c4">x</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c7">Q</font> implies <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c8">P1</font>, <font color="Maroon" title="c9">Q1</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">s</font><span class="kw"> such that </span><br/><a NAME="E2:62"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
( <font color="Maroon" title="c4">x</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c8">P1</font> &amp; <font color="Maroon" title="c5">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c9">Q1</font> )
 <span class="kw">and </span><br/><a NAME="E3:62"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">s</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_1.html#D5" target="_self" title="TOPALG_1:def.5">Def5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:62"/>
( <font color="Maroon" title="c4">x</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<font color="Maroon" title="c7">Q</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="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span></span><br/>

<a NAME="E5:62"/><span class="kw">then </span>
( <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c8">P1</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  &amp; <font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c9">Q1</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="#E2:62"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="topalg_1.html#T47" target="_self" title="TOPALG_1:th.47">Th47</a></span>;<br/>
<a NAME="E6:62"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c8">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c9">Q1</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="ref" href="borsuk_6.html#T83" title="BORSUK_6:th.83">BORSUK_6:83</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:62"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</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">S</font>,<font color="Maroon" title="c2">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E38">A2</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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