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<b>let </b><font color="Maroon">F1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:19"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E2:19_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E3:19_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 ;<br/><a NAME="E4:19_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>hence </b><a NAME="E5:19_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">n</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 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="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E4:19"/>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:19"/><b>then </b><i><font color="Green">E33</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; 0 <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">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:19"/>
( <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</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">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:19"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&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">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E8:19"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<b>hence </b><a NAME="E9:19"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <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">F1</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>


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