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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="matrlin.html#V2" target="_self">finite-dimensional</a> <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>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</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>6</sub></font> be   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>;<br/>









<b>assume </b><a NAME="E1:33"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:33"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</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>3</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="E3:33"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</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>5</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E4:33"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <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>5</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></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>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM8">FinSequence</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 href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:33"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E4:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a></i>
;<br/>
<a NAME="E7:33"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:33"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</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>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E9:33"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</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="relat_1.html#K1">dom</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></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:33_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</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="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><a NAME="E2:33_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="matrlin.html#E7:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E3:33_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:33_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</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/><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E9:33"><i><font color="Green">E30</font></i></a>, <a class="txt" href="matrlin.html#E1:33_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</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>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:33"><i><font color="Green">E27</font></i></a>, <a class="txt" href="matrlin.html#E9:33"><i><font color="Green">E30</font></i></a>, <a class="txt" href="matrlin.html#E1:33_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</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>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E4:33_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</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> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:33_2"><i><font color="Green">E31</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>5</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E4:33"><i><font color="Green">E28</font></i></a>, <a class="txt" href="matrlin.html#E4:33_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E7:33"><i><font color="Green">E29</font></i></a>, <a class="txt" href="matrlin.html#E1:33_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:33"/>
<font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="matrlin.html#E7:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


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