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<span class="kw">let </span><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">pathwise_connected</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E1:98"/>
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#D3" onmouseover="rs('borsuk_2/D3')" onmouseout="rh()">BORSUK_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:98"/>
( <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 class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T75" target="_self" onmouseover="rs('borsuk_6/T75')" onmouseout="rh()">Th75</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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