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<span class="kw">let </span><font color="Maroon" title="c4">S</font>, <font color="Maroon" title="c5">T</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">s1</font>, <font color="Olive" title="b2">s2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">S</font><br/> for <font color="Olive" title="b3">t1</font>, <font color="Olive" title="b4">t2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font><br/> for <font color="Olive" title="b5">l1</font>, <font color="Olive" title="b6">l2</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">s1</font>,<font color="Olive" title="b3">t1</font></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="Olive" title="b2">s2</font>,<font color="Olive" title="b4">t2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span><br/> for <font color="Olive" title="b7">p</font>, <font color="Olive" title="b8">q</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b3">t1</font>,<font color="Olive" title="b4">t2</font>  st <font color="Olive" title="b7">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="Olive" title="b5">l1</font> &amp; <font color="Olive" title="b8">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="Olive" title="b6">l2</font> &amp; <font color="Olive" title="b5">l1</font>,<font color="Olive" title="b6">l2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  holds <br/><font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/><span class="kw">let </span><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>; <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">t1</font>, <font color="Olive" title="b2">t2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font><br/> for <font color="Olive" title="b3">l1</font>, <font color="Olive" title="b4">l2</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Olive" title="b1">t1</font></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="c7">s2</font>,<font color="Olive" title="b2">t2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span><br/> for <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">q</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>  st <font color="Olive" title="b5">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="Olive" title="b3">l1</font> &amp; <font color="Olive" title="b6">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="Olive" title="b4">l2</font> &amp; <font color="Olive" title="b3">l1</font>,<font color="Olive" title="b4">l2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  holds <br/><font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/><span class="kw">let </span><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>; <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">l1</font>, <font color="Olive" title="b2">l2</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></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="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <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>  st <font color="Olive" title="b3">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="Olive" title="b1">l1</font> &amp; <font color="Olive" title="b4">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="Olive" title="b2">l2</font> &amp; <font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  holds <br/><font color="Olive" title="b3">p</font>,<font color="Olive" title="b4">q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/><span class="kw">let </span><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#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></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="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></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="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>  st <font color="Olive" title="b1">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="Olive" title="b2">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> &amp; <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>  holds <br/><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/><span class="kw">let </span><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>; <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="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> &amp; <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>  implies <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )</span><br/>

















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
( <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> )
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<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> 
 ; <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="c12">p</font>,<font color="Maroon" title="c13">q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/>

<span class="kw">consider </span><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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> ,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E4:31"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
( <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#K18" title="BORSUK_1:func.18">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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K18" title="BORSUK_1:func.18">I[01]</a>  holds <br/> ( <font color="Maroon" title="c14">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</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#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> ) ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a></span>;<br/>
<span class="kw">take </span>
 <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c14">f</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</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_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">K569</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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; <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive">b<sub>1</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>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t2</font> ) ) )</span><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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></span>;<br/>
<span class="kw">hence </span><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">K569</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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; <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive">b<sub>1</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>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t2</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="topalg_4.html#E22"><span class="lab"><font color="Green" title="E20">Lm4</font></span></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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