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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Olive" title="b2">F</font> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Olive" title="b1">F</font> )</span><br/>



<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E3:47"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 ) </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E4:47"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
 <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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c3">S1</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c4">S2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan_a.html#D7" target="_self" title="JORDAN_A:def.7">Def7</a></span>;<br/>
<span class="kw">take </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c5">F</font> )</span><br/>

<span class="kw">set </span><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/>
<span class="kw">thus </span><a NAME="E5:47"/>
<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> 
  <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c5">F</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:47_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="47_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:47_1_1_1"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:47_1_1_1"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">and </span><br/><a NAME="E5:47_1_1_1"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:47_1_1_1"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:47_1_1_1"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/><a NAME="E8:47_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E5:47_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E6:47_1_1_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E7:47_1_1_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="jordan_a.html#T25" target="_self" title="JORDAN_A:th.25">Th25</a></span>;<br/><span class="kw">hence </span><a NAME="E9:47_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E4:47_1_1_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E5:47_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E6:47_1_1_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:47_1_1_2"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:47_1_1_2"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">and </span><br/><a NAME="E5:47_1_1_2"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E38">A2</font></span></a></span>;<br/><a NAME="E6:47_1_1_2"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_1_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><a NAME="E7:47_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E5:47_1_1_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="jordan_a.html#T26" target="_self" title="JORDAN_A:th.26">Th26</a></span>;<br/><span class="kw">hence </span><a NAME="E8:47_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E4:47_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E6:47_1_1_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:47"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c5">F</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:47_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:47_2"/><span class="lab"><font color="Green" title="E40">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:47_2"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">and </span><br/><a NAME="E5:47_2"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:47_2"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:47_2"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
 <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="E8:47_2"/><span class="lab"><font color="Green" title="E45">A19</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_2"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E6:47_2"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_2"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_2_1_1"/><span class="lab"><font color="Green" title="E46">A20</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:47_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_2"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E5:47_2"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E7:47_2"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E1:47_2_1_1"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="jordan_a.html#T28" target="_self" title="JORDAN_A:th.28">Th28</a></span>;<br/><a NAME="E3:47_2_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:47_2"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E4:47_2"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E5:47_2"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E1:47_2_1_1"><span class="lab"><font color="Green" title="E46">A20</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_2_1_2"/><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:47_2_1_2"/><span class="kw">then </span>
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_2"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="jordan_a.html#T30" target="_self" title="JORDAN_A:th.30">Th30</a></span>;<br/><a NAME="E3:47_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A22</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_2"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:47_2_1_2"/><span class="lab"><font color="Green" title="E48">A23</font></span>: 
<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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_2"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><a NAME="E5:47_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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_2"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E1:47_2_1_2"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="ref" href="jordan_a.html#T32" target="_self" title="JORDAN_A:th.32">Th32</a></span>;<br/><a NAME="E6:47_2_1_2"/><span class="kw">then </span>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_2_1_2"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E7:47_2_1_2"/><span class="kw">then </span>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E3:47_2"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E1:47_2_1_2"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="txt" href="#E3:47_2_1_2"><span class="lab"><font color="Green" title="E47">A22</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:47_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:47"/>
 <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_2.html#K2" title="XXREAL_2:func.2">min</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="xxreal_2.html#T68" title="XXREAL_2:th.68">XXREAL_2:68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
