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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#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="c6">s1</font>, <font color="Maroon" title="c7">s2</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="c8">t1</font>, <font color="Maroon" title="c9">t2</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="c10">l1</font>, <font color="Maroon" title="c11">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="c6">s1</font>,<font color="Maroon" title="c8">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="c9">t2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>;<br/><b>let </b><font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c8">t1</font>,<font color="Maroon" title="c9">t2</font>;<br/>

















<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c12">p</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="c10">l1</font> &amp; <font color="Maroon" title="c13">q</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="c11">l2</font> )
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c10">l1</font>,<font color="Maroon" title="c11">l2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c14">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E4:31"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c14">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; ( for <font color="Olive" title="b1">a</font> being  <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>  holds <br/> ( <font color="Maroon" title="c14">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">a</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c14">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">a</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <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>  holds <br/> ( <font color="Maroon" title="c14">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c8">t1</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> &amp; <font color="Maroon" title="c14">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c9">t2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> ) ) ) ) )
 <b>by </b><i><a class="txt" href="topalg_4.html#E2:31"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a></i>;<br/>
<b>take </b>
 <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c14">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a><br/>

<a NAME="E5:31"/>
<font color="Maroon" title="c14">f</font> is    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c10">l1</font>,<font color="Maroon" title="c11">l2</font>
 
<b>by </b><i><a class="txt" href="topalg_4.html#E2:31"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="topalg_4.html#E4:31"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
(  <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c14">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">K567</a>()) holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">K567</a>()) holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t1</font> &amp; <span class="p1">(<span class="default"><a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t2</font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="topalg_4.html#E1:31"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="topalg_4.html#E2:31"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="topalg_4.html#E22"><i><font color="Green" title="E20">Lm4</font></i></a></i>;<br/>


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