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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</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">c</font> being   <a href="topalg_6.html#V7" title="TOPALG_6:attr.7">with_endpoints</a>  <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Olive" title="b1">c</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Olive" title="b1">c</font>, <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Olive" title="b1">c</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</font> be   <a href="topalg_6.html#V7" title="TOPALG_6:attr.7">with_endpoints</a>  <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</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"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies <font color="Maroon" title="c2">c</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c</font>, <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</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"> <font color="Maroon" title="c2">c</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c</font>, <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">t1</font> =  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">t2</font> =  <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c2">c</font> as   <a href="topalg_6.html#V4" title="TOPALG_6:attr.4">parametrized-curve</a>  <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T23" target="_self" onmouseover="rs('topalg_6/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">S</font> being    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> , <font color="Maroon" title="c7">p</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:57"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c7">p</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#D4" target="_self" onmouseover="rs('topalg_6/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">p</font> as   <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>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:57"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="topmetr.html#T19" onmouseover="rs('topmetr/T19')" onmouseout="rh()">TOPMETR:19</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/>
<a NAME="E7:57"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="xxreal_2.html#T25" onmouseover="rs('xxreal_2/T25')" onmouseout="rh()">XXREAL_2:25</a></span>;<br/>
<a NAME="E8:57"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="xxreal_2.html#T29" onmouseover="rs('xxreal_2/T29')" onmouseout="rh()">XXREAL_2:29</a></span>;<br/>
<a NAME="E9:57"/><span class="kw">then </span>
 <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c</font>, <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E7:57"><span class="lab"><font color="Green" title="E35">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:57"/>
<font color="Maroon" title="c2">c</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c</font>, <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E7:57"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E8:57"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</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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