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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>













<i><font color="Green">E33</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_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="E1:44_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</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"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:44_2"/><b>then </b><i><font color="Green">E35</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"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</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/><a NAME="E3:44_2"/>
<font color="Maroon">c<sub>8</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="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:44_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E4:44_2"/><b>then </b><i><font color="Green">E36</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>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_1_1"/>
( <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> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="matrlin.html#E4:44_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_1_1"><b>suppose </b></a><a NAME="E1:44_2_1_1_1"/><i><font color="Green">E37</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> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:44_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E3:44_2_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</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/><b>thus </b><span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E2:44_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</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>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E2:44_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:44_2_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:44_2_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrlin.html#E3:44_2_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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#E1:44_2_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrlin.html#E3:44_2_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <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#E1:44_2_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrlin.html#E3:44_2_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_1_2"><b>suppose </b></a><a NAME="E1:44_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:44_2_1_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/><a NAME="E4:44_2_1_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E5:44_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E6:44_2_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E2:44_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</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>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E2:44_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E3:44_2_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E3:44_2_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrlin.html#E6:44_2_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>7</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#E3:44_2_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrlin.html#E6:44_2_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <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#E3:44_2_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrlin.html#E4:44_2_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="matrlin.html#E6:44_2_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:44_2"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:44"/>
 <a href="matrlin.html#K8" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:44"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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