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<b>let </b><font color="Maroon" title="c1">X</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">a1</font>, <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">c1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b4">P1</font>, <font color="Olive" title="b5">P2</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font><br/> for <font color="Olive" title="b6">Q1</font>, <font color="Olive" title="b7">Q2</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">c1</font>  st <font color="Olive" title="b4">P1</font>,<font color="Olive" title="b5">P2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Olive" title="b6">Q1</font>,<font color="Olive" title="b7">Q2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  holds <br/><font color="Olive" title="b4">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b6">Q1</font>,<font color="Olive" title="b5">P2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b7">Q2</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">a1</font>, <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">c1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</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">P1</font>, <font color="Olive" title="b2">P2</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">b1</font><br/> for <font color="Olive" title="b3">Q1</font>, <font color="Olive" title="b4">Q2</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">c1</font>  st <font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">P2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Olive" title="b3">Q1</font>,<font color="Olive" title="b4">Q2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  holds <br/><font color="Olive" title="b1">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b3">Q1</font>,<font color="Olive" title="b2">P2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b4">Q2</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">P1</font>, <font color="Maroon" title="c6">P2</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">b1</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">Q1</font>, <font color="Olive" title="b2">Q2</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">c1</font>  st <font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">P2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Olive" title="b1">Q1</font>,<font color="Olive" title="b2">Q2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  holds <br/><font color="Maroon" title="c5">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b1">Q1</font>,<font color="Maroon" title="c6">P2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b2">Q2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> </span><br/><b>let </b><font color="Maroon" title="c7">Q1</font>, <font color="Maroon" title="c8">Q2</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">c1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">P2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c7">Q1</font>,<font color="Maroon" title="c8">Q2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  implies <font color="Maroon" title="c5">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q1</font>,<font color="Maroon" title="c6">P2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">Q2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  )</span><br/>











<a NAME="E1:100"/>
( <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">b1</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">c1</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="E2:100"/>
( <font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">P2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c7">Q1</font>,<font color="Maroon" title="c8">Q2</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  implies <font color="Maroon" title="c5">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q1</font>,<font color="Maroon" title="c6">P2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">Q2</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#T83" target="_self" title="BORSUK_6:th.83">Th83</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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