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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">c1</font>, <font color="Olive" title="b2">c2</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 <font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a>  holds <br/>(  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Olive" title="b1">c1</font> <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="Olive" title="b2">c2</font> &amp;  <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Olive" title="b1">c1</font> <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="Olive" title="b2">c2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">c1</font>, <font color="Maroon" title="c3">c2</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"> ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a>  implies (  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font> <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="c3">c2</font> &amp;  <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c1</font> <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="c3">c2</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:66"/>
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</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"> (  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font> <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="c3">c2</font> &amp;  <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c1</font> <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="c3">c2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font><span class="kw"> such that </span><br/><a NAME="E3:66"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
( <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )
 ;<br/>
<a NAME="E4:66"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> &amp; 1 is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E6:66"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
( <font color="Maroon" title="c8">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b1">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> holds <br/> ( <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E38">A1</font></span></a></span>;<br/>
<a NAME="E7:66"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
( <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E39">A2</font></span></a></span>;<br/>
<a NAME="E8:66"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
( <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E39">A2</font></span></a></span>;<br/>
<a NAME="E9:66"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</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> &amp; 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E44">A7</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>
;<br/>
<span class="lab"><font color="Green" title="E45">A8</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>)</span>)</span> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>
;<br/>
<a NAME="E12:66"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T40" onmouseover="rs('xxreal_2/T40')" onmouseout="rh()">XXREAL_2:40</a></span>;<br/>
<a NAME="E13:66"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T40" onmouseover="rs('xxreal_2/T40')" onmouseout="rh()">XXREAL_2:40</a></span>;<br/>
<span class="lab"><font color="Green" title="E48">A11</font></span>: <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:66"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span>

;<br/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:66"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span>

;<br/>
<a NAME="E16:66"/><span class="kw">then </span>
<font color="Maroon" title="c6">p1</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> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E9:66"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E10:66"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:66"/>
 <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font> <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="c3">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E14:66"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="txt" href="#E9:66"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E11:66"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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">  <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c1</font> <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="c3">c2</font></span><br/>

<span class="lab"><font color="Green" title="E49">A12</font></span>: <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:66"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span>

;<br/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:66"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span>

;<br/>
<a NAME="E20:66"/><span class="kw">then </span>
<font color="Maroon" title="c6">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E9:66"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E10:66"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:66"/>
 <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c1</font> <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="c3">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:66"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E18:66"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E9:66"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E11:66"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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/>


</div>
