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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>consider </b><font color="Maroon">c<sub>4</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><b> such that </b><br/><a NAME="E2:2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E3:2"/><i><font color="Green">E4</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:2"/><i><font color="Green">E5</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D10">RLVECT_2:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</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><b> such that </b><br/><a NAME="E6:2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E7:2"/><i><font color="Green">E7</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E8:2"/><i><font color="Green">E8</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D10">RLVECT_2:def 10</a></i>;<br/>
<b>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><b> such that </b><br/><a NAME="E10:2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E11:2"/><i><font color="Green">E10</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E12:2"/><i><font color="Green">E11</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D10">RLVECT_2:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E14:2"/><i><font color="Green">E12</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E15:2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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="txt" href="rlvect_3.html#E14:2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E17:2"/><i><font color="Green">E14</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:2_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:2_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E17:2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:2_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/><a NAME="E4:2_1"/>
<font color="Maroon">c<sub>10</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>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E1:2_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E14:2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:2_1"/><b>then </b>
not <font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E3:2_1"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E15</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E17:2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rlvect_2.html#T5">RLVECT_2:5</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/>
<i><font color="Green">E16</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E1"><i><font color="Green">Lemma1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E19:2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E22:2"/><i><font color="Green">E17</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E23:2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</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="txt" href="rlvect_3.html#E22:2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E25:2"/><i><font color="Green">E19</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4"><b>now </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="E1:2_4"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:2_4"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E25:2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:2_4"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/><a NAME="E4:2_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E1:2_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:2_4"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E22:2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:2_4"/><b>then </b>
not <font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:2_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E3:2_4"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E20</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> = 
0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E25:2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="rlvect_2.html#T5">RLVECT_2:5</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>;<br/>
<i><font color="Green">E21</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E1"><i><font color="Green">Lemma1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E27:2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E30:2"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K2">rng</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"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E31:2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</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="txt" href="rlvect_3.html#E30:2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E33:2"/><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:2_7"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><a NAME="E2:2_7"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E33:2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:2_7"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</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>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/><a NAME="E4:2_7"/>
<font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E1:2_7"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:2_7"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E30:2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_7"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:2_7"/><b>then </b>
not <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:2_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E3:2_7"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E25</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> = 
0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E33:2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rlvect_2.html#T5">RLVECT_2:5</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>;<br/>
<i><font color="Green">E26</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E1"><i><font color="Green">Lemma1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E35:2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<a NAME="E37:2"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_10"/>
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:2_10"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/>
<a NAME="E4:2_10"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E7:2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rlvect_3.html#E14:2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rlvect_3.html#E2:2_10"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:2_10"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:2"/><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_11"/>
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:2_11"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>;<br/>
<a NAME="E4:2_11"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E11:2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rlvect_3.html#E22:2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rlvect_3.html#E2:2_11"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:2_11"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_12"/>
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>

<a NAME="E2:2_12"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>;<br/>
<a NAME="E4:2_12"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E3:2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="rlvect_3.html#E30:2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rlvect_3.html#E2:2_12"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:2_12"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E40:2"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E2:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="rlvect_3.html#E6:2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rlvect_3.html#E10:2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="rlvect_3.html#E15:2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rlvect_3.html#E23:2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rlvect_3.html#E31:2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="rlvect_3.html#E37:2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="rlvect_3.html#E38:2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E41:2"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E42:2"/><b>then </b><i><font color="Green">E30</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E3:2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="rlvect_3.html#E7:2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rlvect_3.html#E11:2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rlvect_3.html#E14:2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rlvect_3.html#E22:2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rlvect_3.html#E30:2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E43:2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</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="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E44:2"/><b>then </b>
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E45:2"/><b>then </b><i><font color="Green">E31</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E42:2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E46:2"/><b>then </b><i><font color="Green">E32</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#T65">FINSEQ_1:65</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E48:2"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E49:2"/><i><font color="Green">E34</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"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E50:2"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</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="E1:2_13"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_13"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E4:2_13"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</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>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="rlvect_3.html#E3:2_13"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:2_13"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E48:2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rlvect_3.html#E3:2_13"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:2_13"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:2_13"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</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>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E45:2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:2_13"/><b>then </b>
( <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#R2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_3.html#E49:2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rlvect_3.html#E6:2_13"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/>
<a NAME="E10:2_13"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:2_13"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E4:2_13"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:2_14"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> implies ( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_14_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:2_14_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E48:2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:2_14_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E4:2_14_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E50:2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:2_14_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E6:2_14_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:2_14"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> )
 ;<br/><a NAME="E3:2_14"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:2_14"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E48:2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_15"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:2_15"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</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>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E45:2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_15"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:2_15"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#R2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/><a NAME="E5:2_15"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_15"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E49:2"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E4:2_15"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>
;<br/><b>hence </b><a NAME="E7:2_15"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E53:2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E51:2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>
<a NAME="E54:2"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</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="E1:2_16"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/>
<a NAME="E2:2_16"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_3.html#E45:2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>then </b><i><font color="Green">E40</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
;<br/>
<a NAME="E4:2_16"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E50:2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E53:2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E4:2_16"><i><font color="Green">E41</font></i></a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E6:2_16"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E1:2_16"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rlvect_3.html#E3:2_16"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E55:2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E50:2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E56:2"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E48:2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E57:2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>23</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E55:2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#T75">FINSEQ_4:75</a></i>;<br/>
<a NAME="E58:2"/>
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>, <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="rlvect_3.html#E50:2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="rlvect_3.html#E56:2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="rlvect_3.html#E55:2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rlvect_3.html#E57:2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/>
<a NAME="E61:2"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/>
<a NAME="E62:2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>25</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_2.html#T51">FINSEQ_2:51</a></i>;<br/>
<a NAME="E64:2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E65:2"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E61:2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_2.html#T9">RLVECT_2:9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E67:2"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E68:2"/><i><font color="Green">E46</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"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E69:2"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</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="E1:2_17"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>29</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_17"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E4:2_17"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</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>30</sub></font> = <font color="Maroon">c<sub>29</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="rlvect_3.html#E3:2_17"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:2_17"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E67:2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rlvect_3.html#E3:2_17"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:2_17"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:2_17"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</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="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E45:2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:2_17"/><b>then </b>
( <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</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_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#R2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_3.html#E68:2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rlvect_3.html#E6:2_17"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/>
<a NAME="E10:2_17"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:2_17"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E4:2_17"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_18"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:2_18"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> implies ( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_18_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:2_18_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E67:2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:2_18_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E4:2_18_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E69:2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:2_18_1"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E6:2_18_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:2_18"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 ;<br/><a NAME="E3:2_18"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:2_18"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E67:2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_19"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_19"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>28</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:2_19"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</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="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E45:2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_19"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:2_19"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#R2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/><a NAME="E5:2_19"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_19"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E68:2"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E4:2_19"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>
;<br/><b>hence </b><a NAME="E7:2_19"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</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_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E72:2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</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_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E70:2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>
<a NAME="E73:2"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</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="E1:2_20"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/>
<a NAME="E2:2_20"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rlvect_3.html#E45:2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>then </b><i><font color="Green">E52</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
;<br/>
<a NAME="E4:2_20"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E69:2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</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_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E72:2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E40:2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E4:2_20"><i><font color="Green">E53</font></i></a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E6:2_20"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E1:2_20"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlvect_3.html#E3:2_20"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E74:2"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E69:2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E75:2"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E67:2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E76:2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>27</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E74:2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#T75">FINSEQ_4:75</a></i>;<br/>
<a NAME="E77:2"/>
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>, <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="rlvect_3.html#E69:2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rlvect_3.html#E75:2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>28</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="rlvect_3.html#E74:2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlvect_3.html#E76:2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/>
<a NAME="E80:2"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/>
<a NAME="E81:2"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>29</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_2.html#T51">FINSEQ_2:51</a></i>;<br/>
<a NAME="E83:2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E84:2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E80:2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rlvect_2.html#T9">RLVECT_2:9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E86:2"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E87:2"/><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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E88:2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>31</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</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="E1:2_21"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>33</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_21"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>31</sub></font>
 <b>and </b><br/><a NAME="E4:2_21"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</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>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="rlvect_3.html#E3:2_21"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:2_21"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:2_21"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</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>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E86:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rlvect_3.html#E87:2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rlvect_3.html#E3:2_21"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:2_21"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</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="rlvect_3.html#E4:2_21"><i><font color="Green">E60</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>31</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/>
<a NAME="E90:2"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E61:2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="rlvect_3.html#E62:2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rlvect_3.html#E86:2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a></i>;<br/>
<a NAME="E91:2"/><b>then </b><i><font color="Green">E60</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E92:2"/><i><font color="Green">E61</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E80:2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="rlvect_3.html#E81:2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a>, <a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_22"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>33</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_22"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/><a NAME="E2:2_22"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E86:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rlvect_3.html#E90:2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="rlvect_3.html#E92:2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="rlvect_3.html#E1:2_22"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><b>set </b><font color="Maroon">c<sub>35</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font>;<br/><a NAME="E4:2_22"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E86:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rlvect_3.html#E92:2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_22"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>then </b><i><font color="Green">E64</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E2:2_22"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>
;<br/><a NAME="E6:2_22"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E86:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rlvect_3.html#E92:2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_22"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>then </b><i><font color="Green">E65</font></i>: <font color="Maroon">c<sub>30</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E2:2_22"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E8:2_22"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E75:2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rlvect_3.html#E86:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_22"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:2_22"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E74:2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlvect_3.html#E75:2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:2_22"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E67:2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="rlvect_3.html#E9:2_22"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E71:2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rlvect_3.html#E2:2_22"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E2:2_22"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E13:2_22"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E10:2_22"><i><font color="Green">E67</font></i></a>, <a class="ref" href="rlvect_2.html#T40">RLVECT_2:40</a></i>;<br/><a NAME="E14:2_22"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</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>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E2:2_22"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E15:2_22"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E46:2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_3.html#E56:2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rlvect_3.html#E86:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_22"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E16:2_22"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E55:2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rlvect_3.html#E56:2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:2_22"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E48:2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="rlvect_3.html#E16:2_22"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E19:2_22"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E86:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rlvect_3.html#E1:2_22"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E52:2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="rlvect_3.html#E2:2_22"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="rlvect_3.html#E2:2_22"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E17:2_22"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rlvect_2.html#T40">RLVECT_2:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><b>hence </b><a NAME="E22:2_22"/>
<font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E87:2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rlvect_3.html#E5:2_22"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rlvect_3.html#E7:2_22"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rlvect_3.html#E13:2_22"><i><font color="Green">E68</font></i></a>, <a class="txt" href="rlvect_3.html#E14:2_22"><i><font color="Green">E69</font></i></a>, <a class="txt" href="rlvect_3.html#E19:2_22"><i><font color="Green">E72</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E94:2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E91:2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E95:2"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E92:2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E96:2"/>
 <a href="rlvect_2.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E4:2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rlvect_3.html#E8:2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rlvect_3.html#E12:2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="rlvect_3.html#E20:2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rlvect_3.html#E28:2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rlvect_3.html#E36:2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="rlvect_3.html#E65:2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="rlvect_3.html#E84:2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rlvect_3.html#E86:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rlvect_3.html#E87:2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rlvect_3.html#E92:2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="rlvect_3.html#E94:2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rlvect_2.html#T4">RLVECT_2:4</a></i>;<br/>


</div>
