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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="E1:79"/><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> &amp; <font color="Maroon">FT</font> is <a href="fin_topo.html#V2">symmetric</a> )
 ;<br/>

<a NAME="E2:79"/><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> )
 
<b>by </b><i><a class="ref" href="fintopo6.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E3:79"/>
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/>; <i><font color="Red">:: according to </font></i><a class="ref" href="fintopo6.html#D9" target="_self">FINTOPO6:def 9</a><br/>

<b>let </b><font color="Maroon">i2</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j2</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>;<br/>



<b>assume </b><a NAME="E4:79"/><i><font color="Green">E33</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:79_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j2</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1"/>
( <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font> or <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> )
 </span><b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <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="79_1_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">n1</font> = <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1;<br/><b>set </b><font color="Maroon">n2</font> = <font color="Maroon">i2</font>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1, <font color="Maroon">n2</font> = <font color="Maroon">i2</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="E3:79_1_1_1"/>
<font color="Maroon">i2</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:79_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:79_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E6:79_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>reconsider </b><font color="Maroon">g2</font> = <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</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">n1</font></span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> ;<br/><a NAME="E8:79_1_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E9:79_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:79_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E11:79_1_1_1"/>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E12:79_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <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="ref" href="fintopo6.html#D3" target="_self">Def3</a>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E13:79_1_1_1"/><i><font color="Green">E81</font></i>: 
 <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">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>then </b><i><font color="Green">E82</font></i>: <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>
;<br/><i><font color="Green">E83</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> = 
<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">n2</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">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, </i>
;<br/><a NAME="E16:79_1_1_1"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E17:79_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font>;<br/><a NAME="E18:79_1_1_1"/><i><font color="Green">E86</font></i>: 
 <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">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_1_3_1"/>
( <font color="Maroon">n1</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">n1</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="79_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_1"/>
<font color="Maroon">n1</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"><a NAME="E2:79_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">n1</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="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:79_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T15">XREAL_1:15</a></i>;<br/><a NAME="E4:79_1_1_1_3_1_1"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font> <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">n1</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="fintopo6.html#E6:79_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:79_1_1_1_3_1_1"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E6:79_1_1_1"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E6:79_1_1_1_3_1_1"/><i><font color="Green">E99</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="txt" href="fintopo6.html#E10:79_1_1_1"><i><font color="Green">E60</font></i></a></i>
<br/>.= 
<font color="Maroon">x2</font>
<b>by </b><i>, , <a class="txt" href="fintopo6.html#E10:79_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, </i>
;<br/><a NAME="E8:79_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">g2</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">g2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E9:79_1_1_1_3_1_1"/><i><font color="Green">E101</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</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="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span></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">n1</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="E10:79_1_1_1_3_1_1"/><i><font color="Green">E102</font></i>: 
 <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">n2</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#T21">FINSEQ_5:21</a></i>;<br/><a NAME="E11:79_1_1_1_3_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">n1</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="E12:79_1_1_1_3_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> <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="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E13:79_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> <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="E14:79_1_1_1_3_1_1"/>
<font color="Maroon">g2</font> is <a href="fintopo6.html#V2" target="_self">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_3_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:79_1_1_1_3_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="fintopo6.html#T8" target="_self">Th8</a>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fintopo6.html#D6" target="_self">FINTOPO6:def 6</a><br/>

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">z1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>;<br/>



<b>assume </b><a NAME="E2:79_1_1_1_3_1_1_2"/><i><font color="Green">E106</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E3:79_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</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="E5:79_1_1_1_3_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:79_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="ref" href="fintopo6.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:79_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_1_3_1_1_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:79_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">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>
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><i><font color="Green">E111</font></i>: <font color="Maroon">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a>, , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a>, </i>
;<br/><a NAME="E5:79_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E112</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E6:79_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">i</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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><font color="Maroon">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a>, </i>
;<br/><b>hence </b><a NAME="E8:79_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">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="Maroon">z1</font>
 <b>by </b><i>, , <a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green">E71</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_1_2_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_1_3_1_1_2_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, </i>;<br/><a NAME="E3:79_1_1_1_3_1_1_2_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:79_1_1_1_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E5:79_1_1_1_3_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:79_1_1_1_3_1_1_2_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 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">n1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a>, , , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E7:79_1_1_1_3_1_1_2_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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">n1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><i><font color="Green">E118</font></i>: <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">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> = 
<font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">i</font>

;<br/><i><font color="Green">E174</font></i>: <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">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n2</font>
<b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1

;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1_2_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_1_3_1_1_2_1_2_3"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n2</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</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="79_1_1_1_3_1_1_2_1_2_3_1"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_1_2_1_2_3_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_1_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E3:79_1_1_1_3_1_1_2_1_2_3_1"/><b>then </b>
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E4:79_1_1_1_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green">E176</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:79_1_1_1_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green">E177</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font> <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">n1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, </i>;<br/><i><font color="Green">E178</font></i>: <font color="Maroon">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a>, , , , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, <a class="ref" href="fintopo6.html#T16" target="_self">Th16</a>, </i>
;<br/><i><font color="Green">E179</font></i>: <font color="Maroon">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a>, <a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green">E71</font></i></a>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E8:79_1_1_1_3_1_1_2_1_2_3_1"/><i><font color="Green">E180</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a>, </i>;<br/><a NAME="E9:79_1_1_1_3_1_1_2_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, </i>;<br/><b>hence </b><a NAME="E10:79_1_1_1_3_1_1_2_1_2_3_1"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">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="Maroon">z1</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_1_2_1_2_3_2"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_1_2_1_2_3_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><i><font color="Green">E182</font></i>: <font color="Maroon">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a>, , , </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>
;<br/><font color="Maroon">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a>, <a class="txt" href="fintopo6.html#E12:79_1_1_1"><i><font color="Green">E71</font></i></a>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><b>hence </b><a NAME="E4:79_1_1_1_3_1_1_2_1_2_3_2"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">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="Maroon">z1</font>
 <b>by </b><i>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E15:79_1_1_1_3_1_1"/><b>then </b>
 <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="Maroon">g2</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self">Th7</a>, <a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="ref" href="fintopo6.html#T10" target="_self">Th10</a>, , </i>;<br/><a NAME="E16:79_1_1_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T15">XREAL_1:15</a>, </i>;<br/><b>hence </b><a NAME="E17:79_1_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T12">XREAL_1:12</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_1_3_1_2"/>
<font color="Maroon">n1</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="E2:79_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E3:79_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>hence </b><a NAME="E4:79_1_1_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E20:79_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1, <font color="Maroon">n2</font> = <font color="Maroon">j2</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="E3:79_1_1_2"/>
<font color="Maroon">j2</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:79_1_1_2"/><b>then </b>
<font color="Maroon">j2</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E10"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:79_1_1_2"/><b>then </b>
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E6:79_1_1_2"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>reconsider </b><font color="Maroon">g2</font> = <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</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">n1</font></span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">FT</font> ;<br/><a NAME="E8:79_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E9:79_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:79_1_1_2"/><b>then </b><i><font color="Green">E185</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E11:79_1_1_2"/>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E12:79_1_1_2"/><b>then </b><i><font color="Green">E186</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <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="ref" href="fintopo6.html#D3" target="_self">Def3</a>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E13:79_1_1_2"/><i><font color="Green">E187</font></i>: 
 <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">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>then </b><i><font color="Green">E188</font></i>: <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>
;<br/><i><font color="Green">E189</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> = 
<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">n2</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">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, </i>
;<br/><a NAME="E16:79_1_1_2"/><i><font color="Green">E190</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E17:79_1_1_2"/><b>then </b><i><font color="Green">E191</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font>;<br/><a NAME="E18:79_1_1_2"/><i><font color="Green">E192</font></i>: 
 <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">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_2_3_1"/>
( <font color="Maroon">n1</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">n1</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="79_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_1"/>
<font color="Maroon">n1</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"><a NAME="E2:79_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon">n1</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="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:79_1_1_2_3_1_1"/><b>then </b><i><font color="Green">E193</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T15">XREAL_1:15</a></i>;<br/><a NAME="E4:79_1_1_2_3_1_1"/><i><font color="Green">E194</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font> <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">n1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, , </i>;<br/><a NAME="E5:79_1_1_2_3_1_1"/><i><font color="Green">E195</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><a NAME="E6:79_1_1_2_3_1_1"/><i><font color="Green">E196</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font></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="rfinseq.html#D2">RFINSEQ:def 2</a>, </i>
;<br/><a NAME="E8:79_1_1_2_3_1_1"/><b>then </b><i><font color="Green">E197</font></i>: 
<font color="Maroon">g2</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">g2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E9:79_1_1_2_3_1_1"/><i><font color="Green">E198</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</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="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span></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">n1</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="E10:79_1_1_2_3_1_1"/><i><font color="Green">E199</font></i>: 
 <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">n2</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#T21">FINSEQ_5:21</a></i>;<br/><a NAME="E11:79_1_1_2_3_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">n1</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="E12:79_1_1_2_3_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> <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="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E13:79_1_1_2_3_1_1"/><b>then </b><i><font color="Green">E200</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> <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="E14:79_1_1_2_3_1_1"/>
<font color="Maroon">g2</font> is <a href="fintopo6.html#V2" target="_self">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2_3_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:79_1_1_2_3_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="fintopo6.html#T8" target="_self">Th8</a>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fintopo6.html#D6" target="_self">FINTOPO6:def 6</a><br/>

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">z1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font>;<br/>



<b>assume </b><a NAME="E2:79_1_1_2_3_1_1_2"/><i><font color="Green">E201</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E3:79_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E202</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</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="E5:79_1_1_2_3_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:79_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="fintopo6.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E7:79_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_2_3_1_1_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_1_2_1_1"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_2_3_1_1_2_1_1"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:79_1_1_2_3_1_1_2_1_1"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">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>
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><i><font color="Green">E206</font></i>: <font color="Maroon">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a>, , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a>, </i>
;<br/><a NAME="E5:79_1_1_2_3_1_1_2_1_1"/><i><font color="Green">E207</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E6:79_1_1_2_3_1_1_2_1_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">i</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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><font color="Maroon">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a>, </i>
;<br/><b>hence </b><a NAME="E8:79_1_1_2_3_1_1_2_1_1"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">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="Maroon">z1</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_1_2_1_2"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_2_3_1_1_2_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, </i>;<br/><a NAME="E3:79_1_1_2_3_1_1_2_1_2"/><b>then </b><i><font color="Green">E210</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:79_1_1_2_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E5:79_1_1_2_3_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:79_1_1_2_3_1_1_2_1_2"/><b>then </b><i><font color="Green">E211</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 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">n1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a>, , , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E7:79_1_1_2_3_1_1_2_1_2"/><b>then </b><i><font color="Green">E212</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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">n1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><i><font color="Green">E213</font></i>: <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">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> = 
<font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a>, </i>
<br/>.= 
<font color="Maroon">i</font>

;<br/><i><font color="Green">E214</font></i>: <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">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n2</font>
<b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1

;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1_2_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_2_3_1_1_2_1_2_3"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n2</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</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="79_1_1_2_3_1_1_2_1_2_3_1"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_1_2_1_2_3_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><a NAME="E2:79_1_1_2_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green">E215</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E3:79_1_1_2_3_1_1_2_1_2_3_1"/><b>then </b>
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E4:79_1_1_2_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green">E216</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:79_1_1_2_3_1_1_2_1_2_3_1"/><b>then </b><i><font color="Green">E217</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font> <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">n1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><i><font color="Green">E218</font></i>: <font color="Maroon">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a>, , , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, <a class="ref" href="fintopo6.html#T16" target="_self">Th16</a>, </i>
;<br/><i><font color="Green">E219</font></i>: <font color="Maroon">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E8:79_1_1_2_3_1_1_2_1_2_3_1"/><i><font color="Green">E220</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a>, </i>;<br/><a NAME="E9:79_1_1_2_3_1_1_2_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, </i>;<br/><b>hence </b><a NAME="E10:79_1_1_2_3_1_1_2_1_2_3_1"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">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="Maroon">z1</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_1_2_1_2_3_2"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_1_2_1_2_3_2"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><i><font color="Green">E222</font></i>: <font color="Maroon">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a>, , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>
;<br/><font color="Maroon">z2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a></i>
;<br/><b>hence </b><a NAME="E4:79_1_1_2_3_1_1_2_1_2_3_2"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">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="Maroon">z1</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self">Th7</a>, <a class="ref" href="fin_topo.html#D15">FIN_TOPO:def 15</a>, , <a class="ref" href="fintopo6.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E15:79_1_1_2_3_1_1"/><b>then </b>
 <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="Maroon">g2</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self">Th7</a>, <a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="ref" href="fintopo6.html#T10" target="_self">Th10</a>, , </i>;<br/><a NAME="E16:79_1_1_2_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T15">XREAL_1:15</a>, </i>;<br/><b>hence </b><a NAME="E17:79_1_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T12">XREAL_1:12</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:79_1_1_2_3_1_2"/>
<font color="Maroon">n1</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="E2:79_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E3:79_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>hence </b><a NAME="E4:79_1_1_2_3_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="fintopo6.html#D2" target="_self">Def2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E20:79_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


</div>
