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<b>let </b><font color="Maroon">FT</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/><b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>, <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>;<br/>







<b>assume </b><a NAME="E1:76"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">g</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 ;<br/>

<a NAME="E2:76"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">g</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; ( for <font color="Olive">h</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font>  st <font color="Olive">h</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">h</font> ) )
 
<b>by </b><i><a class="ref" href="fintopo6.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:76"/><b>then </b><i><font color="Green">E33</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x11</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; <font color="Olive">x11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">x11</font> ) )
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E4:76"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:76"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="txt" href="fintopo6.html#E4:76"><i><font color="Green">E41</font></i></a>, <a class="txt" href="fintopo6.html#E6:76"><i><font color="Green">E43</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_1_1_1"/>
not <font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Maroon">x1</font>,<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 ;<br/><b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E3:76_1_1_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">h</font> is <a href="fintopo6.html#V2" target="_self">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> ;<br/><a NAME="E5:76_1_1_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:76_1_1_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>;<br/><a NAME="E7:76_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:76_1_1_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:76_1_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:76_1_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T23">JORDAN3:23</a></i>;<br/><a NAME="E11:76_1_1_1_1"/><i><font color="Green">E83</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:76_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:76_1_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E14:76_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:76_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:76_1_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E17:76_1_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font> is <a href="fintopo6.html#V2" target="_self">continuous</a>
 <b>by </b><i><a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</font></i></a>, <a class="txt" href="fintopo6.html#E6:76"><i><font color="Green">E43</font></i></a>, </i>;<br/><a NAME="E18:76_1_1_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">s</font> is <a href="fintopo6.html#V2" target="_self">continuous</a>
 <b>by </b><i>, <a class="txt" href="fintopo6.html#E3:76"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E19:76_1_1_1_1"/><i><font color="Green">E97</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E20:76_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T36">FINSEQ_5:36</a></i>;<br/><a NAME="E21:76_1_1_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E22:76_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E23:76_1_1_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E24:76_1_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E25:76_1_1_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E26:76_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="jordan4.html#T18">JORDAN4:18</a>, <a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</font></i></a></i>
;<br/><b>hence </b><a NAME="E28:76_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="fintopo6.html#D3" target="_self">Def3</a>, <a class="txt" href="fintopo6.html#E6:76"><i><font color="Green">E43</font></i></a>, <a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="txt" href="fintopo6.html#E1:76_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="fintopo6.html#E3:76_1_1_1_1"><i><font color="Green">E60</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:76_1_1_1"/>
<font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Maroon">x1</font>,<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:76_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><a NAME="E3:76_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:76_1_1_2"/>
<font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Maroon">x1</font>,<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="fintopo6.html#E6:76"><i><font color="Green">E43</font></i></a>, <a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:76"/>
<font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="fintopo6.html#R3" target="_self">is_minimum_path_in</a> <font color="Maroon">A</font>,<font color="Maroon">x1</font>,<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 ;<br/>


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