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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">S</font> being   <a href="jordan_a.html#M1" title="JORDAN_A:mode.1">Segmentation</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b2">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Olive" title="b1">S</font>,<font color="Olive" title="b4">j</font></span>)</span></span>)</span> where <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</font> <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="Olive" title="b1">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Olive" title="b1">S</font>,<font color="Olive" title="b4">j</font> ) </span>}</span>  &amp;  <a href="jordan_a.html#K5" title="JORDAN_A:func.5">S-Gap</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Olive" title="b2">F</font> )</span><br/><b>let </b><font color="Maroon" title="c2">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">  ex <font color="Olive" title="b1">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b3">j</font></span>)</span></span>)</span> where <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b3">j</font> ) </span>}</span>  &amp;  <a href="jordan_a.html#K5" title="JORDAN_A:func.5">S-Gap</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Olive" title="b1">F</font> )</span><br/>



<b>consider </b><font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c4">S2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:46"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c3">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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="c2">S</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>  ) </span>}</span> 
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c4">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:46"/><i><font color="Green" title="E39">A3</font></i>: 
 <a href="jordan_a.html#K5" title="JORDAN_A:func.5">S-Gap</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c3">S1</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c4">S2</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan_a.html#D7" target="_self" title="JORDAN_A:def.7">Def7</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c3">S1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">S2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span>  &amp;  <a href="jordan_a.html#K5" title="JORDAN_A:func.5">S-Gap</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c5">F</font> )</span><br/>

<b>set </b><font color="Maroon" title="c6">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E5:46"/>
<font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font> &amp;  <a href="jordan_a.html#K5" title="JORDAN_A:func.5">S-Gap</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c5">F</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> or <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:46_1"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1"/>
( <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S1</font> or <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S2</font> )
 </span><b>by </b><i><a class="txt" href="jordan_a.html#E1:46_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span> </span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">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> <b> such that </b><br/><a NAME="E3:46_1_1_1"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 <b>and </b><br/><a NAME="E4:46_1_1_1"/><i><font color="Green" title="E42">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">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="c2">S</font> )
 <b>and </b><br/><a NAME="E5:46_1_1_1"/><i><font color="Green" title="E43">A7</font></i>: 
not <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a> 
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:46"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/><a NAME="E6:46_1_1_1"/>
 <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c8">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:46_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="jordan_a.html#E5:46_1_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="jordan_a.html#T25" target="_self" title="JORDAN_A:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E7:46_1_1_1"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:46_1_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="jordan_a.html#E4:46_1_1_1"><i><font color="Green" title="E42">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span> </span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">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> <b> such that </b><br/><a NAME="E3:46_1_1_2"/><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c8">j</font></span>)</span>
 <b>and </b><br/><a NAME="E4:46_1_1_2"/><i><font color="Green" title="E42">A9</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</font>
 <b>and </b><br/><a NAME="E5:46_1_1_2"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:46"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/><a NAME="E6:46_1_1_2"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c8">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="c2">S</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E5:46_1_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E7:46_1_1_2"/>
 <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:46_1_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="jordan_a.html#E5:46_1_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="jordan_a.html#T26" target="_self" title="JORDAN_A:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E8:46_1_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:46_1_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="jordan_a.html#E4:46_1_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="jordan_a.html#E6:46_1_1_2"><i><font color="Green" title="E44">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E6:46"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</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#K5" title="JORDAN_A:func.5">S-Gap</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c5">F</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span>  or <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> )</span><br/>

<b>assume </b><a NAME="E1:46_2"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : ( 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#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="c2">S</font> &amp;  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b2">j</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">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> <b> such that </b><br/><a NAME="E3:46_2"/><i><font color="Green" title="E40">A12</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1k.html#K4" title="JORDAN1K:func.4">dist_min</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 <b>and </b><br/><a NAME="E4:46_2"/><i><font color="Green" title="E41">A13</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <b>and </b><br/><a NAME="E5:46_2"/><i><font color="Green" title="E42">A14</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font>
 <b>and </b><br/><a NAME="E6:46_2"/><i><font color="Green" title="E43">A15</font></i>: 
<font color="Maroon" title="c9">j</font> <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="c2">S</font>
 <b>and </b><br/><a NAME="E7:46_2"/><i><font color="Green" title="E44">A16</font></i>: 
 <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c8">i</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c9">j</font>
 ;<br/>
<a NAME="E8:46_2"/><i><font color="Green" title="E45">A17</font></i>: 
<font color="Maroon" title="c8">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="c2">S</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E5:46_2"><i><font color="Green" title="E42">A14</font></i></a>, <a class="txt" href="jordan_a.html#E6:46_2"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_2_1"/>
( <font color="Maroon" title="c9">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="c2">S</font> or <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font> )
 </span><b>by </b><i><a class="txt" href="jordan_a.html#E6:46_2"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1"><b>suppose </b></a><a NAME="E1:46_2_1_1"/><i><font color="Green" title="E46">A18</font></i>: 
<font color="Maroon" title="c9">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="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font></span><br/><div class="add"><a NAME="E2:46_2_1_1"/>
not <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a> 
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:46_2"><i><font color="Green" title="E41">A13</font></i></a>, <a class="txt" href="jordan_a.html#E5:46_2"><i><font color="Green" title="E42">A14</font></i></a>, <a class="txt" href="jordan_a.html#E7:46_2"><i><font color="Green" title="E44">A16</font></i></a>, <a class="txt" href="jordan_a.html#E1:46_2_1_1"><i><font color="Green" title="E46">A18</font></i></a>, <a class="ref" href="jordan_a.html#T28" target="_self" title="JORDAN_A:th.28">Th28</a></i>;<br/><a NAME="E3:46_2_1_1"/><b>then </b>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S1</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:46"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="jordan_a.html#E3:46_2"><i><font color="Green" title="E40">A12</font></i></a>, <a class="txt" href="jordan_a.html#E4:46_2"><i><font color="Green" title="E41">A13</font></i></a>, <a class="txt" href="jordan_a.html#E5:46_2"><i><font color="Green" title="E42">A14</font></i></a>, <a class="txt" href="jordan_a.html#E1:46_2_1_1"><i><font color="Green" title="E46">A18</font></i></a></i>;<br/><b>hence </b><a NAME="E4:46_2_1_1"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0: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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_2"><b>suppose </b></a><a NAME="E1:46_2_1_2"/><i><font color="Green" title="E46">A19</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font></span><br/><div class="add"><a NAME="E2:46_2_1_2"/><b>then </b>
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E7:46_2"><i><font color="Green" title="E44">A16</font></i></a>, <a class="ref" href="jordan_a.html#T30" target="_self" title="JORDAN_A:th.30">Th30</a></i>;<br/><a NAME="E3:46_2_1_2"/><b>then </b><i><font color="Green" title="E47">A20</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:46_2"><i><font color="Green" title="E41">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:46_2_1_2"/><i><font color="Green" title="E48">A21</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan_a.html#E8:46_2"><i><font color="Green" title="E45">A17</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E5:46_2_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan_a.html#E7:46_2"><i><font color="Green" title="E44">A16</font></i></a>, <a class="txt" href="jordan_a.html#E1:46_2_1_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="jordan_a.html#T32" target="_self" title="JORDAN_A:th.32">Th32</a></i>;<br/><a NAME="E6:46_2_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:46_2_1_2"><i><font color="Green" title="E48">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E7:46_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S2</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:46"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="jordan_a.html#E3:46_2"><i><font color="Green" title="E40">A12</font></i></a>, <a class="txt" href="jordan_a.html#E1:46_2_1_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="jordan_a.html#E3:46_2_1_2"><i><font color="Green" title="E47">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E8:46_2_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0: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/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E7:46"/>
 <a href="jordan_a.html#K5" title="JORDAN_A:func.5">S-Gap</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:46"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="jordan_a.html#T1" target="_self" title="JORDAN_A:th.1">Th1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
