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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">M1</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">k</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">M2</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">m</font>,<font color="Maroon">k</font>, <a href="numbers.html#K1">REAL</a> ;<br/>









<i><font color="Green">E43</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</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="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M2</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="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">i</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="matrprob.html#K3" target="_self">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:34_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</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="E3:34_2"/>
<font color="Maroon">i</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">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:34_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</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">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</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="34_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_2_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M1</font> or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">n</font> ) )
 </span><b>by </b><i>, <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="34_2_1_1_1"><b>suppose </b></a><a NAME="E1:34_2_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M1</font>
 ;<br/><div class="add"><a NAME="E2:34_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:34_2_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</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="matrprob.html#K3" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <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="34_2_1_1_2"><b>suppose </b></a><a NAME="E1:34_2_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:34_2_1_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E4:34_2_1_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:34_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:34_2_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M2</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="matrprob.html#K3" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , <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="E6:34_2"/>
<span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
 <a href="matrprob.html#K3" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10">^</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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