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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font>;<br/>









<b>assume </b><a NAME="E1:11"/><i><font color="Green">E10</font></i>: 
( ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="gobrd10.html#R2" target="_self">are_adjacent2</a>  holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> ) ) )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:11"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<a NAME="E4:11"/><i><font color="Green">E12</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="gobrd10.html#E3:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:11"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E6:11"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>thus </b><a NAME="E7:11"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:11_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:11_1"/><b>then </b><i><font color="Green">E14</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:11_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:11_1"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1"><b>case </b></a><a NAME="E1:11_1_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd10.html#E3:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_2"><b>case </b></a><a NAME="E1:11_1_1_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:11_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E3:11_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <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="E4:11_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E5:11_1_1_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E6:11_1_1_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 )
 <b>by </b><i><a class="txt" href="gobrd10.html#E4:11"><i><font color="Green">E12</font></i></a>, <a class="txt" href="gobrd10.html#E6:11"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:11_1_1_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 )
 <b>by </b><i><a class="txt" href="gobrd10.html#E2:11_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="gobrd10.html#E4:11_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="gobrd10.html#E5:11_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:11_1_1_1_2"/><i><font color="Green">E18</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 ) ) &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1 &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1 &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd10.html#R2" target="_self">are_adjacent2</a>  ) )
 <b>by </b><i><a class="txt" href="gobrd10.html#E6:11_1_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="gobrd10.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E10:11_1_1_1_2"/><i><font color="Green">E19</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E11:11_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:11_1_1_1_2"/><i><font color="Green">E20</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 ) )
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E14:11_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:11_1_1_1_2"/><i><font color="Green">E21</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 ) )
 ;<br/><a NAME="E17:11_1_1_1_2"/><i><font color="Green">E22</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd10.html#E10:11_1_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E18:11_1_1_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:11_1_1_1_2"/>
1 <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">c<sub>10</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E10:11_1_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E20:11_1_1_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E21:11_1_1_1_2"/><i><font color="Green">E25</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E17:11_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E22:11_1_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E18:11_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><i><font color="Green">E27</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="gobrd10.html#E21:11_1_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E17:11_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="gobrd10.html#E22:11_1_1_1_2"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="gobrd10.html#E6:11"><i><font color="Green">E13</font></i></a></i>
;<br/><a NAME="E24:11_1_1_1_2"/><i><font color="Green">E28</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E17:11_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E25:11_1_1_1_2"/><i><font color="Green">E29</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gobrd10.html#E17:11_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></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">c<sub>12</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> );<br/><a NAME="E26:11_1_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E24:11_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="gobrd10.html#E25:11_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gobrd10.html#E17:11_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="gobrd10.html#E26:11_1_1_1_2"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="gobrd10.html#E6:11"><i><font color="Green">E13</font></i></a></i>
;<br/><a NAME="E28:11_1_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="txt" href="gobrd10.html#E3:11"><i><font color="Green">E11</font></i></a>, <a class="txt" href="gobrd10.html#E23:11_1_1_1_2"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E29:11_1_1_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11_1_1_1_2_3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>14</sub></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">c<sub>12</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:11_1_1_1_2_3"/><i><font color="Green">E34</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1_1_2_3_1_1"/>
( <font color="Maroon">c<sub>14</sub></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">c<sub>12</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_2_3_1_1_1"><b>case </b></a><a NAME="E1:11_1_1_1_2_3_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>12</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1_2_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_2_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1_1_2_3_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>12</sub></font> or <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_2_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:11_1_1_1_2_3_1_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd10.html#E2:11_1_1_1_2_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:11_1_1_1_2_3_1_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E3:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E6:11_1_1_1_2_3_1_1_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gobrd10.html#E3:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:11_1_1_1_2_3_1_1_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E3:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E43</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="gobrd10.html#E4:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E3:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E38</font></i></a></i>
;<br/><a NAME="E9:11_1_1_1_2_3_1_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd10.html#E5:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd10.html#E7:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E5:11_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E5:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E11:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:11_1_1_1_2_3_1_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E13:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E8:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd10.html#E11:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><i><font color="Green">E47</font></i>: <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="gobrd10.html#E6:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gobrd10.html#E3:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E38</font></i></a></i>
;<br/><a NAME="E15:11_1_1_1_2_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E5:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd10.html#E9:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E16:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E5:11_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E17:11_1_1_1_2_3_1_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E18:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E14:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E19:11_1_1_1_2_3_1_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="E20:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd10.html#E1:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E21:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E22:11_1_1_1_2_3_1_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E20:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E23:11_1_1_1_2_3_1_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gobrd10.html#E20:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E24:11_1_1_1_2_3_1_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E20:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E55</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="gobrd10.html#E21:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E20:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E50</font></i></a></i>
;<br/><a NAME="E26:11_1_1_1_2_3_1_1_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="gobrd10.html#E22:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E27:11_1_1_1_2_3_1_1_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="gobrd10.html#E24:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E28:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E5:11_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E22:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd10.html#E26:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E29:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E30:11_1_1_1_2_3_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gobrd10.html#E20:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E31:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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/><a NAME="E32:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gobrd10.html#E20:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E33:11_1_1_1_2_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E22:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd10.html#E26:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="gobrd10.html#E27:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E34:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E5:11_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E35:11_1_1_1_2_3_1_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E36:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E25:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd10.html#E29:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><i><font color="Green">E62</font></i>: <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="gobrd10.html#E23:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gobrd10.html#E20:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E50</font></i></a></i>
;<br/><a NAME="E38:11_1_1_1_2_3_1_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E39:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E32:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="gobrd10.html#E34:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E40:11_1_1_1_2_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd10.html#R2" target="_self">are_adjacent2</a> 
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E2:11_1_1_1_2_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="gobrd10.html#E2:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E41:11_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd10.html#R2" target="_self">are_adjacent2</a> 
 <b>by </b><i><a class="txt" href="gobrd10.html#E8:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd10.html#E14:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="gobrd10.html#E25:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd10.html#E37:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="gobrd10.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E42:11_1_1_1_2_3_1_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>12</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd10.html#E1:11"><i><font color="Green">E10</font></i></a>, <a class="txt" href="gobrd10.html#E1:11_1_1_1_2_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd10.html#E1:11_1_1_1_2_3_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd10.html#E18:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd10.html#E36:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd10.html#E39:11_1_1_1_2_3_1_1_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_2_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:11_1_1_1_2_3_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_1_2_3_1_1_1_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>12</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:11_1_1_1_2_3_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>12</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_2_3_1_1_2"><b>case </b></a><a NAME="E1:11_1_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_1_2_3_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>12</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><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="E4:11_1_1_1_2_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</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> <font color="Maroon">c<sub>12</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E30:11_1_1_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="gobrd10.html#E28:11_1_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="gobrd10.html#E29:11_1_1_1_2"><i><font color="Green">E32</font></i></a>);<br/></i><a NAME="E31:11_1_1_1_2"/><i><font color="Green">E34</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">c<sub>10</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E20:11_1_1_1_2"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E32:11_1_1_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd10.html#E10:11_1_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E33:11_1_1_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E34:11_1_1_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E32:11_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E35:11_1_1_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E32:11_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E36:11_1_1_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gobrd10.html#E32:11_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E40</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<font color="Maroon">c<sub>12</sub></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">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="gobrd10.html#E35:11_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E32:11_1_1_1_2"><i><font color="Green">E35</font></i></a></i>
;<br/><a NAME="E38:11_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E33:11_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E39:11_1_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E4:11_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="gobrd10.html#E37:11_1_1_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><i><font color="Green">E42</font></i>: <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<font color="Maroon">c<sub>13</sub></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">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="gobrd10.html#E36:11_1_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E16:11_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gobrd10.html#E32:11_1_1_1_2"><i><font color="Green">E35</font></i></a></i>
;<br/><a NAME="E41:11_1_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="gobrd10.html#E9:11_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd10.html#E34:11_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E42:11_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="gobrd10.html#E4:11_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="gobrd10.html#E13:11_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gobrd10.html#E30:11_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd10.html#E31:11_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="gobrd10.html#E39:11_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="gobrd10.html#E40:11_1_1_1_2"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:11_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


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


</div>
