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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>;<br/><b>let </b><font color="Maroon" title="c2">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:44"/>
<font color="Maroon" title="c2">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:44"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c3">h</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="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E4:44"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c3">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:44"/><i><font color="Green" title="E37">A3</font></i>: 
8 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 <b>and </b><br/><a NAME="E6:44"/><i><font color="Green" title="E38">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E7:44"/><i><font color="Green" title="E39">A5</font></i>: 
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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">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">h</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E8:44"/><i><font color="Green" title="E40">A6</font></i>: 
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">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">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">h</font> &amp; <font color="Olive" title="b2">W</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">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">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> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
 <b>and </b><br/><a NAME="E9:44"/><i><font color="Green" title="E41">A7</font></i>: 
for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">W</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">h</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">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">C</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
 <b>and </b><br/><a NAME="E10:44"/><i><font color="Green" title="E42">A8</font></i>: 
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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">h</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">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></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E11:44"/><i><font color="Green" title="E43">A9</font></i>: 
<span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E12:44"/><i><font color="Green" title="E44">A10</font></i>: 
<span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E13:44"/><i><font color="Green" title="E45">A11</font></i>: 
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</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">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E14:44"/><i><font color="Green" title="E46">A12</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</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">h</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">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> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</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>
 <b>and </b><br/><a NAME="E15:44"/><i><font color="Green" title="E47">A13</font></i>: 
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>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">h</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</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">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">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>
 <b>by </b><i><a class="ref" href="jordan7.html#T21" title="JORDAN7:th.21">JORDAN7:21</a></i>;<br/>
<a NAME="E16:44"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan_a.html#E5:44"><i><font color="Green" title="E37">A3</font></i></a></i>;<br/>
<a NAME="E17:44"/><b>then </b>
<font color="Maroon" title="c3">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E18:44"/><b>then </b>
1 <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">h</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E19:44"/><b>then </b>
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E3:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">h</font> = <font color="Maroon" title="c3">h</font> as    <a href="jordan_a.html#M1" title="JORDAN_A:mode.1">Segmentation</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="jordan_a.html#E4:44"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="jordan_a.html#E5:44"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="jordan_a.html#E6:44"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="jordan_a.html#E7:44"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="jordan_a.html#E10:44"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="jordan_a.html#E11:44"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="jordan_a.html#E12:44"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="jordan_a.html#E13:44"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="jordan_a.html#E14:44"><i><font color="Green" title="E46">A12</font></i></a>, <a class="txt" href="jordan_a.html#E15:44"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="jordan_a.html#D3" target="_self" title="JORDAN_A:def.3">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">h</font>
;<br/>

<a NAME="E21:44"/>
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>   holds  <a href="jordan_a.html#K3" title="JORDAN_A:func.3">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c4">h</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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> ;<br/>

<b>consider </b><font color="Maroon" title="c6">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span><b> such that </b><br/><a NAME="E2:44_1"/><i><font color="Green" title="E48">A14</font></i>: 
<font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">i</font>
 <b>and </b><br/><a NAME="E3:44_1"/><i><font color="Green" title="E49">A15</font></i>: 
 <a href="jordan_a.html#K3" title="JORDAN_A:func.3">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c6">W</font>
 <b>by </b><i><a class="ref" href="jordan_a.html#D5" target="_self" title="JORDAN_A:def.5">Def5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1"/>
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">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="c4">h</font> ) or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> or  not <font color="Maroon" title="c5">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="c4">h</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1"/><i><font color="Green" title="E50">A16</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">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="c4">h</font> )
 ;<br/><div class="add"><a NAME="E2:44_1_1_1"/><b>then </b>
 <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">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="c4">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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>
 <b>by </b><i><a class="ref" href="jordan_a.html#D4" target="_self" title="JORDAN_A:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E3:44_1_1_1"/>
 <a href="jordan_a.html#K3" title="JORDAN_A:func.3">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E8:44"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="jordan_a.html#E2:44_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="jordan_a.html#E3:44_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="jordan_a.html#E1:44_1_1_1"><i><font color="Green" title="E50">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_2"/>
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> or  not <font color="Maroon" title="c5">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="c4">h</font> )
 ;<br/><div class="add"><a NAME="E2:44_1_1_2"/><b>then </b>
 <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">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="c4">h</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="c4">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</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="ref" href="jordan_a.html#D4" target="_self" title="JORDAN_A:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E3:44_1_1_2"/>
 <a href="jordan_a.html#K3" title="JORDAN_A:func.3">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E9:44"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="jordan_a.html#E2:44_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="jordan_a.html#E3:44_1"><i><font color="Green" title="E49">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E22:44"/>
 <a href="jordan_a.html#K4" title="JORDAN_A:func.4">diameter</a> <font color="Maroon" title="c4">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
 <b>by </b><i><a class="ref" href="jordan_a.html#T34" target="_self" title="JORDAN_A:th.34">Th34</a></i>;<br/>


</div>
