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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">G'</font>, <font color="Olive" title="b3">G1</font>, <font color="Olive" title="b4">G2</font> being  <a href="matrixj1.html#NM2" title="MATRIXJ1:NM.2">FinSequence_of_Matrix</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">G'</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Olive" title="b3">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Olive" title="b4">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Olive" title="b2">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Olive" title="b4">G2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">G'</font>, <font color="Maroon" title="c4">G1</font>, <font color="Maroon" title="c5">G2</font> be   <a href="matrixj1.html#NM2" title="MATRIXJ1:NM.2">FinSequence_of_Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G'</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span> )</span><br/>









<b>assume </b><a NAME="E1:105"/><i><font color="Green" title="E68">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c6">GG'</font> = <font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font>;<br/>
<b>set </b><font color="Maroon" title="c7">G12</font> = <font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font>;<br/>
<b>set </b><font color="Maroon" title="c8">GG1</font> = <font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font>;<br/>
<b>set </b><font color="Maroon" title="c9">GG2</font> = <font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font>;<br/>
<a NAME="E2:105"/><i><font color="Green" title="E69">A2</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">G1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">G'</font> )
 
<b>by </b><i><a class="txt" href="#E1:105"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="matrixj1.html#D10" target="_self" title="MATRIXJ1:def.10">Def10</a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:105"/><b>then </b><i><font color="Green" title="E70">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">G1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E4:105"/><i><font color="Green" title="E71">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">G1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:105"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E5:105"/><b>then </b><i><font color="Green" title="E72">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E3:105"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">G1</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> )</span><br/><b>assume </b><a NAME="E1:105_1"/><i><font color="Green" title="E73">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">G1</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font></span><br/><a NAME="E2:105_1"/><i><font color="Green" title="E74">A7</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="#E4:105"><i><font color="Green" title="E71">A4</font></i></a>, <a class="txt" href="#E1:105_1"><i><font color="Green" title="E73">A6</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:105_1"/><b>then </b><i><font color="Green" title="E75">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrixj1.html#D10" target="_self" title="MATRIXJ1:def.10">Def10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_1_1"/>
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> or  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="#E2:105_1"><i><font color="Green" title="E74">A7</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_1"><b>suppose </b></a><a NAME="E1:105_1_1_1_1"/><i><font color="Green" title="E76">A9</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:105_1"><i><font color="Green" title="E75">A8</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="#E1:105_1_1_1_1"><i><font color="Green" title="E76">A9</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font>
<b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="#E1:105_1_1_1_1"><i><font color="Green" title="E76">A9</font></i></a>, <a class="ref" href="matrixj1.html#D10" target="_self" title="MATRIXJ1:def.10">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font>
<b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="#E1:105_1_1_1_1"><i><font color="Green" title="E76">A9</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_2"><b>suppose </b></a><a NAME="E1:105_1_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_1_1_2"/><i><font color="Green" title="E76">A10</font></i>: 
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_1_1_2_1_1"/>
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G2</font> or  not <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:105_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G2</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c11">n</font></span><br/><div class="add"><b>hence </b><a NAME="E2:105_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G2</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="#E1:105"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="#E3:105_1_1_1_2"><i><font color="Green" title="E76">A10</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:105_1_1_1_2_1_1_2"/><i><font color="Green" title="E77">A11</font></i>: 
not <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G2</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c11">n</font></span><br/><div class="add"><a NAME="E2:105_1_1_1_2_1_1_2"/><b>then </b>
( ( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G2</font> ) &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> )
 <b>by </b><i><a class="txt" href="#E3:105_1_1_1_2"><i><font color="Green" title="E76">A10</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:105_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:105_1_1_1_2"><i><font color="Green" title="E76">A10</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E4:105_1_1_1_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:105"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> = 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c5">G2</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c11">n</font>
<b>by </b><i><a class="txt" href="#E1:105_1_1_1_2_1_1_2"><i><font color="Green" title="E77">A11</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G2</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:105_1"><i><font color="Green" title="E75">A8</font></i></a>, <a class="txt" href="#E3:105_1_1_1_2"><i><font color="Green" title="E76">A10</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c11">n</font>
<b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="#E3:105_1_1_1_2"><i><font color="Green" title="E76">A10</font></i></a>, <a class="ref" href="matrixj1.html#D10" target="_self" title="MATRIXJ1:def.10">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font>
<b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="#E3:105"><i><font color="Green" title="E70">A3</font></i></a>, <a class="txt" href="#E3:105_1_1_1_2"><i><font color="Green" title="E76">A10</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:105_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c10">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:105"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G'</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c3">G'</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(+)</a> <font color="Maroon" title="c5">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:105"><i><font color="Green" title="E70">A3</font></i></a>, <a class="txt" href="#E5:105"><i><font color="Green" title="E72">A5</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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