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<b>let </b><font color="Maroon">S</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">T</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">s1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">t1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">t2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">l1</font> be    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">t1</font></span><a href="borsuk_1.html#K7">]</a></span>,<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">t2</font></span><a href="borsuk_1.html#K7">]</a></span>;<br/><b>let </b><font color="Maroon">l2</font> be    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">t1</font></span><a href="borsuk_1.html#K7">]</a></span>,<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">t2</font></span><a href="borsuk_1.html#K7">]</a></span>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">l1</font>,<font color="Maroon">l2</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">s1</font>,<font color="Maroon">s2</font>, <font color="Maroon">q</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">s1</font>,<font color="Maroon">s2</font>;<br/>



















<b>assume </b><b>that </b><br/><a NAME="E1:28"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a>  <a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">q</font> <a href="funct_2.html#R2">=</a>  <a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">l2</font> )
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">l1</font>,<font color="Maroon">l2</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 ;<br/>

<b>thus </b><a NAME="E3:28"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a><br/>

<b>thus </b><a NAME="E1:28_1"/>
(  <a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">K539</a>()) holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#M1">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">K539</a>()) holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> ) ) ) ) )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:28"/>
(  <a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">K539</a>()) holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#M1">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">K539</a>()) holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> ) ) ) ) )
 <b>by </b><i>, , </i>;<br/>


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