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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</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">a</font>,<font color="Olive" title="b2">b</font><br/> for <font color="Olive" title="b4">f</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>   st <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b3">P</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> </span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">a</font>,<font color="Maroon" title="c3">b</font><br/> for <font color="Olive" title="b2">f</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>   st <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">P</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> </span><br/><b>let </b><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">a</font>,<font color="Maroon" title="c3">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>   st <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> </span><br/><b>let </b><font color="Maroon" title="c5">f</font> be  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  )</span><br/>







<a NAME="E1:59"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D3" title="BORSUK_2:def.3">BORSUK_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:59"/>
( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  )
 <b>by </b><i><a class="ref" href="borsuk_6.html#T53" target="_self" title="BORSUK_6:th.53">Th53</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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