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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</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">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">S</font> being   <a href="jordan_a.html#M1" title="JORDAN_A:mode.1">Segmentation</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">S</font> holds <br/> <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font></span><br/><b>let </b><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">S</font> being   <a href="jordan_a.html#M1" title="JORDAN_A:mode.1">Segmentation</a> of <font color="Maroon" title="c1">C</font>  st <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">S</font> holds <br/> <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Olive" title="b1">S</font>,<font color="Maroon" title="c2">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font></span><br/><b>let </b><font color="Maroon" title="c3">S</font> be    <a href="jordan_a.html#M1" title="JORDAN_A:mode.1">Segmentation</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">S</font> implies  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c2">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> )</span><br/>





<b>assume </b><a NAME="E1:29"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c2">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font></span><br/>

<a NAME="E2:29"/><b>then </b><i><font color="Green" title="E24">A1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:29"/>
( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">S</font> or <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">S</font> )
 
;<br/>
<a NAME="E4:29"/><b>then </b>
(  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">C</font> or  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">C</font> )
 
<b>by </b><i><a class="txt" href="jordan_a.html#E2:29"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan_a.html#D4" target="_self" title="JORDAN_A:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E5:29"/>
 <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c2">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T10" title="JORDAN16:th.10">JORDAN16:10</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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