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<a NAME="E1:70_1_1"/>
<font color="Maroon" title="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font> is <a href="matrixj1.html#V2" title="MATRIXJ1:attr.2">Square-Matrix-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="matrixj1.html#D6" target="_self" title="MATRIXJ1:def.6">MATRIXJ1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> implies  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">D</font> )</span><br/>

<b>assume </b><a NAME="E1:70_1_1_1"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c4">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="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">D</font></span><br/>

<b>take </b><font color="Maroon" title="c5">L</font> =  <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">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c4">i</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"><font color="Maroon" title="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c1">D</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1_1_1"/>
( <font color="Maroon" title="c4">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">S1</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="c3">S2</font> &amp; <font color="Maroon" title="c4">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">S1</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="#E1:70_1_1_1"><i><font color="Green" title="E49">A1</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="70_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_1_1_1"/>
<font color="Maroon" title="c4">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">S1</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">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c1">D</font></span><br/><div class="add"><a NAME="E2:70_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E3:70_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c1">D</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_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="c3">S2</font> &amp; <font color="Maroon" title="c4">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">S1</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"><font color="Maroon" title="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c1">D</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:70_1_1_1_1_2"/><i><font color="Green" title="E50">A2</font></i>: 
( <font color="Maroon" title="c6">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="c3">S2</font> &amp; <font color="Maroon" title="c4">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">S1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font> )
 ;<br/><a NAME="E4:70_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="#E3:70_1_1_1_1_2"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:70_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c1">D</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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:70_1_1"/>
<font color="Maroon" title="c2">S1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">S2</font> is   <a href="matrixj1.html#NM3" title="MATRIXJ1:NM.3">FinSequence_of_Square-Matrix</a> of <font color="Maroon" title="c1">D</font>
 ; <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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