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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>;<br/>









<b>assume </b><a NAME="E1:3"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E2:3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</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="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E3:3"/><b>then </b><i><font color="Green">E4</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</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>2</sub></font> <b>by </b><i><a class="ref" href="finseq_2.html#T51">FINSEQ_2:51</a></i>;<br/>
<a NAME="E5:3"/><i><font color="Green">E5</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="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E6:3"/><i><font color="Green">E6</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E1:3"><i><font color="Green">E3</font></i></a>, <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a></i>;<br/>
<a NAME="E7:3"/><b>then </b><i><font color="Green">E7</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E5:3"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:3"/><i><font color="Green">E8</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E6:3"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E9:3"/><i><font color="Green">E9</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E5:3"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E10:3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</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="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E3:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a></i>;<br/>
<a NAME="E11:3"/><b>then </b><i><font color="Green">E10</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a>, <a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:3_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E2:3_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><a NAME="E3:3_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E7:3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="vectsp_9.html#E9:3"><i><font color="Green">E9</font></i></a>, <a class="txt" href="vectsp_9.html#E1:3_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="vectsp_9.html#E7:3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="vectsp_9.html#E8:3"><i><font color="Green">E8</font></i></a>, <a class="txt" href="vectsp_9.html#E9:3"><i><font color="Green">E9</font></i></a>, <a class="txt" href="vectsp_9.html#E1:3_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E1:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="vectsp_9.html#E7:3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="vectsp_9.html#E8:3"><i><font color="Green">E8</font></i></a>, <a class="txt" href="vectsp_9.html#E9:3"><i><font color="Green">E9</font></i></a>, <a class="txt" href="vectsp_9.html#E1:3_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="vectsp_9.html#E3:3_1"><i><font color="Green">E12</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"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<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>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E1:3_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E9:3"><i><font color="Green">E9</font></i></a>, <a class="txt" href="vectsp_9.html#E3:3_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="vectsp_9.html#E7:3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="vectsp_9.html#E11:3"><i><font color="Green">E10</font></i></a>, <a class="txt" href="vectsp_9.html#E1:3_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><b>hence </b><a NAME="E7:3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E13:3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E7:3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="vectsp_9.html#E11:3"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
<b>hence </b><a NAME="E14:3"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E3:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="rlvect_2.html#T9">RLVECT_2:9</a></i>;<br/>


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