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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#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:()"><i><font color="Red">::  thesis: </font></i></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>  st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">s1</font>,<font color="Olive" title="b3">t1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>,<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b2">s2</font>,<font color="Olive" title="b4">t2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/>for <font color="Olive" title="b5">L</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">s1</font>,<font color="Olive" title="b3">t1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>,<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b2">s2</font>,<font color="Olive" title="b4">t2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> holds   <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Olive" title="b5">L</font> is    <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></span><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>; <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">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>  st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Olive" title="b1">t1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>,<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Olive" title="b2">t2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/>for <font color="Olive" title="b3">L</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Olive" title="b1">t1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>,<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Olive" title="b2">t2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> holds   <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Olive" title="b3">L</font> is    <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></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>,<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  implies for <font color="Olive" title="b1">L</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>,<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> holds   <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Olive" title="b1">L</font> is    <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> )</span><br/>











<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E17">A1</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>,<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</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">L</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>,<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> holds   <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Olive" title="b1">L</font> is    <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></span><br/>

<b>let </b><font color="Maroon" title="c10">L</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>,<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c10">L</font> is    <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></span><br/>

<b>set </b><font color="Maroon" title="c11">f</font> =  <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c10">L</font>;<br/>
<a NAME="E2:18"/><i><font color="Green" title="E18">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:18"/>
<font color="Maroon" title="c10">L</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/>
<a NAME="E4:18"/><b>then </b><i><font color="Green" title="E19">A3</font></i>: 
 <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c10">L</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="topalg_4.html#T10" target="_self" title="TOPALG_4:th.10">Th10</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green" title="E20">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">L</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>;<br/>
<a NAME="E6:18"/><b>then </b>
<font color="Maroon" title="c2">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">L</font>
 
<b>by </b><i><a class="txt" href="topalg_4.html#E2:18"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E21">A5</font></i>: <span class="p1">(<span class="default"><a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c8">t1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>
<a NAME="E8:18"/>
<font color="Maroon" title="c3">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">L</font>
 
<b>by </b><i><a class="txt" href="topalg_4.html#E2:18"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="topalg_4.html#E5:18"><i><font color="Green" title="E20">A4</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E22">A6</font></i>: <span class="p1">(<span class="default"><a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c9">t2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>
<a NAME="E10:18"/><b>then </b>
<font color="Maroon" title="c8">t1</font>,<font color="Maroon" title="c9">t2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_4.html#E4:18"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="topalg_4.html#E7:18"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:18"/>
 <a href="topalg_4.html#K5" title="TOPALG_4:func.5">pr2</a> <font color="Maroon" title="c10">L</font> is    <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>
 <b>by </b><i><a class="txt" href="topalg_4.html#E4:18"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="topalg_4.html#E7:18"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="topalg_4.html#E9:18"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</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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