<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E1:61"/>
<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="group_4.html#E1:61_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/><a NAME="E3:61_1_1_1_1"/><b>then </b>
(  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:61_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E1:61_1_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_1_2"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/><a NAME="E3:61_1_1_1_2"/><b>then </b>
(  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:61_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E1:61_1_1_1_2"><i><font color="Green">E46</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="E4:61_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:61"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_4.html#T41" target="_self">Th41</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:61_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:61_2"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:61_2"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:61_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3" target="_self">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_4.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E6:61_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>assume </b><a NAME="E1:61_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:61_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:61_2_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="group_4.html#E3:61_2_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E6:61_2_1"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="group_4.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E7:61_2_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="group_4.html#E3:61_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="group_4.html#E3:61_2_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:61_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E10:61_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="group_4.html#E3:61_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="group_4.html#E6:61_2_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E11:61_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="group_4.html#K2" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="group_4.html#E7:61_2_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="E12:61_2_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="group_4.html#E10:61_2_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="group_4.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E13:61_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="group_4.html#E3:61_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="group_4.html#E6:61_2_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E14:61_2_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E16:61_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E17:61_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E18:61_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_2_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> ) );<br/><a NAME="E3:61_2_1_1_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</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">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E81"><i><font color="Green">Lemma38</font></i></a></i>;<br/><a NAME="E4:61_2_1_1_1_1"/><i><font color="Green">E59</font></i>: 
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">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_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="E1:61_2_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="group_4.html#E81"><i><font color="Green">Lemma38</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_1_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E3:61_2_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) &amp; ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="group_4.html#E1:61_2_1_1_1_1_1_1_1_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_1_1_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E3:61_2_1_1_1_1_1_1_1_2"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) &amp; ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="group_4.html#E1:61_2_1_1_1_1_1_1_1_2"><i><font color="Green">E60</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="E3:61_2_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:61_2_1_1_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:61_2_1_1_1_1"/><i><font color="Green">E61</font></i>: 
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">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(<a class="txt" href="group_4.html#E3:61_2_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_1"><i><font color="Green">E59</font></i></a>);<br/></i><a NAME="E8:61_2_1_1_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E6:61_2_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:61_2_1_1_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:61_2_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E4:61_2_1_1_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="group_4.html#E3:61_2_1_1_1_1_2"><i><font color="Green">E64</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_2_1_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="group_4.html#E81"><i><font color="Green">Lemma38</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E6:61_2_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="group_4.html#E7:61_2_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E3:61_2_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_1_2"><i><font color="Green">E65</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E6:61_2_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="group_4.html#E7:61_2_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E3:61_2_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_1_2"><i><font color="Green">E65</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:61_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E10:61_2_1_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 implies for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/> <a href="group_4.html#K3" target="_self">Product</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>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span> );<br/><a NAME="E12:61_2_1_1_1_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st ( 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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2_1_1_1_1_3"/><i><font color="Green">E65</font></i>: 
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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:61_2_1_1_1_1_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E3:61_2_1_1_1_1_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E4:61_2_1_1_1_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_2_1_1_1_1_3_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>18</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="61_2_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_1_3_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/><a NAME="E3:61_2_1_1_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="group_4.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:61_2_1_1_1_1_3_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> 0 <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E5:61_2_1_1_1_1_3_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K3">div</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="group_4.html#E86"><i><font color="Green">Lemma43</font></i></a></i>;<br/><b>hence </b><a NAME="E6:61_2_1_1_1_1_3_1_1_1"/>
 <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E1:61_2_1_1_1_1_3_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="group_4.html#E3:61_2_1_1_1_1_3_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:61_2_1_1_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="group_4.html#E3:61_2_1_1_1_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="group_4.html#E83"><i><font color="Green">Lemma40</font></i></a></i>;<br/><a NAME="E4:61_2_1_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="group_4.html#E2:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:61_2_1_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:61_2_1_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K6">-</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K6">-</a> 2 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E8:61_2_1_1_1_1_3_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E9:61_2_1_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T41">NAT_1:41</a></i>;<br/><a NAME="E10:61_2_1_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E2:61_2_1_1_1_1_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:61_2_1_1_1_1_3_1_1_2"/>
1 <a href="nat_1.html#K2">*</a> 2 <a href="hidden.html#R1">=</a> 2
 ;<br/><a NAME="E12:61_2_1_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="group_4.html#E3:61_2_1_1_1_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="group_4.html#E84"><i><font color="Green">Lemma41</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><a NAME="E14:61_2_1_1_1_1_3_1_1_2"/><i><font color="Green">E74</font></i>: 
 <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E1:61_2_1_1_1_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="group_4.html#E9:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="group_4.html#E10:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="group_4.html#E12:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E15:61_2_1_1_1_1_3_1_1_2"/><i><font color="Green">E75</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E2:61_2_1_1_1_1_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E16:61_2_1_1_1_1_3_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E10:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E17:61_2_1_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E8:61_2_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="group_4.html#E15:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>;<br/><i><font color="Green">E77</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="txt" href="group_4.html#E8:61_2_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="group_4.html#E15:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="rlvect_1.html#K7">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E17:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_1_3_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_2_1_1_1_1_3_1_1_2_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E2:61_2_1_1_1_1_3_1_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E8:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="group_4.html#E17:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T41">NAT_1:41</a></i>;<br/><a NAME="E3:61_2_1_1_1_1_3_1_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:61_2_1_1_1_1_3_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:61_2_1_1_1_1_3_1_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="group_4.html#E1:61_2_1_1_1_1_3_1_1_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E6:61_2_1_1_1_1_3_1_1_2_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_1_3_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_2_1_1_1_1_3_1_1_2_3"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 ;<br/><a NAME="E2:61_2_1_1_1_1_3_1_1_2_3"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,2</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_1_3_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1_3_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="E1:61_2_1_1_1_1_3_1_1_2_3_1_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="group_4.html#E2:61_2_1_1_1_1_3_1_1_2_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1_3_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_1_3_1_1_2_3_1_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_1_3_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:61_2_1_1_1_1_3_1_1_2_3_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="group_4.html#E9:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:61_2_1_1_1_1_3_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>20</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>19</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E17:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:61_2_1_1_1_1_3_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="group_4.html#E17:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="group_4.html#E1:61_2_1_1_1_1_3_1_1_2_3_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E6:61_2_1_1_1_1_3_1_1_2_3_1_1_1"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="group_4.html#E12:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="group_4.html#E85"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E7:61_2_1_1_1_1_3_1_1_2_3_1_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/><b>hence </b><a NAME="E8:61_2_1_1_1_1_3_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E6:61_2_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="group_4.html#E7:61_2_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E2:61_2_1_1_1_1_3_1_1_2_3_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_1_3_1_1_2_3_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="group_4.html#E5:61_2_1_1_1_1_3_1_1_2_3_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="group_4.html#E6:61_2_1_1_1_1_3_1_1_2_3_1_1_1"><i><font color="Green">E84</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1_3_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_1_3_1_1_2_3_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_1_3_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:61_2_1_1_1_1_3_1_1_2_3_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E4:61_2_1_1_1_1_3_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E17:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:61_2_1_1_1_1_3_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="group_4.html#E17:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="group_4.html#E1:61_2_1_1_1_1_3_1_1_2_3_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E6:61_2_1_1_1_1_3_1_1_2_3_1_1_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/><b>hence </b><a NAME="E7:61_2_1_1_1_1_3_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E6:61_2_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="group_4.html#E7:61_2_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E3:61_2_1_1_1_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="group_4.html#E2:61_2_1_1_1_1_3_1_1_2_3_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_1_3_1_1_2_3_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="group_4.html#E5:61_2_1_1_1_1_3_1_1_2_3_1_1_2"><i><font color="Green">E83</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="E4:61_2_1_1_1_1_3_1_1_2_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/></div><b>end;</b></div><a NAME="E21:61_2_1_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="group_4.html#E18:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="group_4.html#E19:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_3.html#T43">FINSEQ_3:43</a></i>;<br/><b>then </b><i><font color="Green">E79</font></i>:  <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" target="_self">Product</a> <span class="p2"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="rlvect_1.html#K7">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E16:61_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="group_4.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E14:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="group_1.html#T49">GROUP_1:49</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K3">div</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K3">div</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E86"><i><font color="Green">Lemma43</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K3">div</a> 2
<b>by </b><i><a class="txt" href="group_4.html#E8:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="group_4.html#E12:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="group_4.html#E87"><i><font color="Green">Lemma44</font></i></a></i>
;<br/><b>hence </b><a NAME="E24:61_2_1_1_1_1_3_1_1_2"/>
 <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E22:61_2_1_1_1_1_3_1_1_2"><i><font color="Green">E79</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="E6:61_2_1_1_1_1_3"/>
 <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E13:61_2_1_1_1_1"/><i><font color="Green">E65</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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>(<a class="txt" href="group_4.html#E12:61_2_1_1_1_1"><i><font color="Green">E64</font></i></a>);<br/></i><a NAME="E14:61_2_1_1_1_1"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="group_4.html#E82"><i><font color="Green">Lemma39</font></i></a></i>;<br/><a NAME="E15:61_2_1_1_1_1"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E86"><i><font color="Green">Lemma43</font></i></a></i>;<br/><a NAME="E16:61_2_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E6:61_2_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E17:61_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T55">FINSEQ_3:55</a></i>;<br/><a NAME="E18:61_2_1_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E12:61_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="group_4.html#E13:61_2_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="group_4.html#E14:61_2_1_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="group_4.html#E15:61_2_1_1_1_1"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E19:61_2_1_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_4.html#E17:61_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="group_4.html#E18:61_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E20:61_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><a NAME="E21:61_2_1_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E9:61_2_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E22:61_2_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp;  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="group_4.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E23:61_2_1_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E21:61_2_1_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E24:61_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E18:61_2_1_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="group_4.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E25:61_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>hence </b><a NAME="E26:61_2_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a>  ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a>  ) );<br/><a NAME="E2:61_2_1_1_1_2"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</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">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E81"><i><font color="Green">Lemma38</font></i></a></i>;<br/><a NAME="E3:61_2_1_1_1_2"/><i><font color="Green">E60</font></i>: 
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">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2_1_1_1_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_2_1_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="group_4.html#E81"><i><font color="Green">Lemma38</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_2_1_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a>  as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E3:61_2_1_1_1_2_1_1_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a>  ) &amp; ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a>  ) )
 <b>by </b><i><a class="txt" href="group_4.html#E1:61_2_1_1_1_2_1_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_2_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a>  as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E3:61_2_1_1_1_2_1_1_1_2"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a>  ) &amp; ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a>  ) )
 <b>by </b><i><a class="txt" href="group_4.html#E1:61_2_1_1_1_2_1_1_1_2"><i><font color="Green">E61</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="E3:61_2_1_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:61_2_1_1_1_2"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:61_2_1_1_1_2"/><i><font color="Green">E62</font></i>: 
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">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(<a class="txt" href="group_4.html#E2:61_2_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="group_4.html#E3:61_2_1_1_1_2"><i><font color="Green">E60</font></i></a>);<br/></i><a NAME="E7:61_2_1_1_1_2"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E5:61_2_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:61_2_1_1_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_2_1_1_1_2_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:61_2_1_1_1_2_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E4:61_2_1_1_1_2_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="group_4.html#E3:61_2_1_1_1_2_2"><i><font color="Green">E65</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_2_1_1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="group_4.html#E81"><i><font color="Green">Lemma38</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E5:61_2_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E6:61_2_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="group_4.html#E3:61_2_1_1_1_2_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_2_2"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_2_1_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E5:61_2_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E6:61_2_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="group_4.html#E3:61_2_1_1_1_2_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_2_2"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_2_1_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:61_2_1_1_1_2_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E9:61_2_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 implies for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/> <a href="group_4.html#K3" target="_self">Product</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K3">"</a>  );<br/><a NAME="E11:61_2_1_1_1_2"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st ( 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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2_1_1_1_2_3"/><i><font color="Green">E66</font></i>: 
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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:61_2_1_1_1_2_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:61_2_1_1_1_2_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E4:61_2_1_1_1_2_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_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="E1:61_2_1_1_1_2_3_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>18</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="61_2_1_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_2_3_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/><a NAME="E3:61_2_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="group_4.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:61_2_1_1_1_2_3_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> 0 <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E5:61_2_1_1_1_2_3_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K3">div</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="group_4.html#E86"><i><font color="Green">Lemma43</font></i></a></i>;<br/><a NAME="E6:61_2_1_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E1:61_2_1_1_1_2_3_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/><b>hence </b><a NAME="E7:61_2_1_1_1_2_3_1_1_1"/>
 <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K3">"</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E3:61_2_1_1_1_2_3_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="group_1.html#T16">GROUP_1:16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:61_2_1_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="group_4.html#E3:61_2_1_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="group_4.html#E83"><i><font color="Green">Lemma40</font></i></a></i>;<br/><a NAME="E4:61_2_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="group_4.html#E2:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:61_2_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:61_2_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K6">-</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K6">-</a> 2 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E8:61_2_1_1_1_2_3_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E9:61_2_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T41">NAT_1:41</a></i>;<br/><a NAME="E10:61_2_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E2:61_2_1_1_1_2_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:61_2_1_1_1_2_3_1_1_2"/>
1 <a href="nat_1.html#K2">*</a> 2 <a href="hidden.html#R1">=</a> 2
 ;<br/><a NAME="E12:61_2_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="group_4.html#E3:61_2_1_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="group_4.html#E84"><i><font color="Green">Lemma41</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><a NAME="E14:61_2_1_1_1_2_3_1_1_2"/><i><font color="Green">E75</font></i>: 
 <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K3">"</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E1:61_2_1_1_1_2_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="group_4.html#E9:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="group_4.html#E10:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="group_4.html#E12:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E15:61_2_1_1_1_2_3_1_1_2"/><i><font color="Green">E76</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E2:61_2_1_1_1_2_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E16:61_2_1_1_1_2_3_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E10:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E17:61_2_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E7:61_2_1_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="group_4.html#E15:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>;<br/><i><font color="Green">E78</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="txt" href="group_4.html#E7:61_2_1_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="group_4.html#E15:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="rlvect_1.html#K7">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E17:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_2_3_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_2_1_1_1_2_3_1_1_2_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E2:61_2_1_1_1_2_3_1_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E8:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="group_4.html#E17:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T41">NAT_1:41</a></i>;<br/><a NAME="E3:61_2_1_1_1_2_3_1_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:61_2_1_1_1_2_3_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:61_2_1_1_1_2_3_1_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="group_4.html#E1:61_2_1_1_1_2_3_1_1_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E6:61_2_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_2_3_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_2_1_1_1_2_3_1_1_2_3"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 ;<br/><a NAME="E2:61_2_1_1_1_2_3_1_1_2_3"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,2</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1_1_2_3_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2_3_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="E1:61_2_1_1_1_2_3_1_1_2_3_1_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="group_4.html#E2:61_2_1_1_1_2_3_1_1_2_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2_3_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_2_3_1_1_2_3_1_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_2_3_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:61_2_1_1_1_2_3_1_1_2_3_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="group_4.html#E9:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:61_2_1_1_1_2_3_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>20</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>19</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E17:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:61_2_1_1_1_2_3_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="group_4.html#E17:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="group_4.html#E1:61_2_1_1_1_2_3_1_1_2_3_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E6:61_2_1_1_1_2_3_1_1_2_3_1_1_1"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="group_4.html#E12:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="group_4.html#E85"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E7:61_2_1_1_1_2_3_1_1_2_3_1_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/><b>hence </b><a NAME="E8:61_2_1_1_1_2_3_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E5:61_2_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E6:61_2_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="group_4.html#E2:61_2_1_1_1_2_3_1_1_2_3_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_2_3_1_1_2_3_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="group_4.html#E5:61_2_1_1_1_2_3_1_1_2_3_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="group_4.html#E6:61_2_1_1_1_2_3_1_1_2_3_1_1_1"><i><font color="Green">E85</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_2_3_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:61_2_1_1_1_2_3_1_1_2_3_1_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1_2_3_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:61_2_1_1_1_2_3_1_1_2_3_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E4:61_2_1_1_1_2_3_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E17:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:61_2_1_1_1_2_3_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="group_4.html#E17:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="group_4.html#E1:61_2_1_1_1_2_3_1_1_2_3_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E6:61_2_1_1_1_2_3_1_1_2_3_1_1_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:61_2_1_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/><b>hence </b><a NAME="E7:61_2_1_1_1_2_3_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E5:61_2_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E6:61_2_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="group_4.html#E3:61_2_1_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="group_4.html#E2:61_2_1_1_1_2_3_1_1_2_3_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="group_4.html#E4:61_2_1_1_1_2_3_1_1_2_3_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="group_4.html#E5:61_2_1_1_1_2_3_1_1_2_3_1_1_2"><i><font color="Green">E84</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="E4:61_2_1_1_1_2_3_1_1_2_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/></div><b>end;</b></div><a NAME="E21:61_2_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="group_4.html#E18:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="group_4.html#E19:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_3.html#T43">FINSEQ_3:43</a></i>;<br/><b>then </b><i><font color="Green">E80</font></i>:  <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" target="_self">Product</a> <span class="p2"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="rlvect_1.html#K7">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="group_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="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E16:61_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="group_1.html#T25">GROUP_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="txt" href="group_4.html#E14:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="group_1.html#T25">GROUP_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#T49">GROUP_1:49</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K3">div</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K3">div</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E86"><i><font color="Green">Lemma43</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K3">div</a> 2
<b>by </b><i><a class="txt" href="group_4.html#E8:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="group_4.html#E12:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="group_4.html#E87"><i><font color="Green">Lemma44</font></i></a></i>
;<br/><b>hence </b><a NAME="E24:61_2_1_1_1_2_3_1_1_2"/>
 <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K3">"</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E22:61_2_1_1_1_2_3_1_1_2"><i><font color="Green">E80</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="E6:61_2_1_1_1_2_3"/>
 <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K3">"</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E12:61_2_1_1_1_2"/><i><font color="Green">E66</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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>(<a class="txt" href="group_4.html#E11:61_2_1_1_1_2"><i><font color="Green">E65</font></i></a>);<br/></i><a NAME="E13:61_2_1_1_1_2"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="group_4.html#E82"><i><font color="Green">Lemma39</font></i></a></i>;<br/><a NAME="E14:61_2_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> 2 <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E86"><i><font color="Green">Lemma43</font></i></a></i>;<br/><a NAME="E15:61_2_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E5:61_2_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E16:61_2_1_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T55">FINSEQ_3:55</a></i>;<br/><a NAME="E17:61_2_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E12:61_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="group_4.html#E1:61_2_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="group_1.html#T56">GROUP_1:56</a></i>;<br/><a NAME="E18:61_2_1_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E12:61_2_1_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="group_4.html#E13:61_2_1_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="group_4.html#E14:61_2_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="group_4.html#E16:61_2_1_1_1_2"><i><font color="Green">E69</font></i></a></i>;<br/><a NAME="E19:61_2_1_1_1_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E17:61_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="group_4.html#E18:61_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="group_2.html#T90">GROUP_2:90</a></i>;<br/><a NAME="E20:61_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E21:61_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><a NAME="E22:61_2_1_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E8:61_2_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E23:61_2_1_1_1_2"/>
( <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp;  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="group_4.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E24:61_2_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E22:61_2_1_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E25:61_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E18:61_2_1_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="group_4.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E26:61_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>hence </b><a NAME="E27:61_2_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E20:61_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:61_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E5:61_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="group_4.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:61"/><b>then </b>
 <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T67">GROUP_2:67</a></i>;<br/>
<b>hence </b><a NAME="E5:61"/>
 <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K7" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E2:61"><i><font color="Green">E45</font></i></a>, <a class="ref" href="group_2.html#T64">GROUP_2:64</a></i>;<br/>


</div>
