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<a NAME="E1:74_2_1"/>
<font color="Maroon" title="c2">S</font> <a href="matrixj1.html#K7" title="MATRIXJ1:func.7">/^</a> <font color="Maroon" title="c3">n</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="74_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">S</font> <a href="matrixj1.html#K7" title="MATRIXJ1:func.7">/^</a> <font color="Maroon" title="c3">n</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">S</font> <a href="matrixj1.html#K7" title="MATRIXJ1:func.7">/^</a> <font color="Maroon" title="c3">n</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/>

<span class="kw">assume </span><a NAME="E1:74_2_1_1"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<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">S</font> <a href="matrixj1.html#K7" title="MATRIXJ1:func.7">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">S</font> <a href="matrixj1.html#K7" title="MATRIXJ1:func.7">/^</a> <font color="Maroon" title="c3">n</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/>

<a NAME="E2:74_2_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">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="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_2_1_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="finseq_5.html#T29" title="FINSEQ_5:th.29">FINSEQ_5:29</a></span>;<br/>
<a NAME="E3:74_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A3</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<span class="kw">take </span><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"><font color="Maroon" title="c2">S</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="matrixj1.html#K7" title="MATRIXJ1:func.7">/^</a> <font color="Maroon" title="c3">n</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/>

<a NAME="E4:74_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="matrixj1.html#K7" title="MATRIXJ1:func.7">/^</a> <font color="Maroon" title="c3">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="matrixj1.html#K7" title="MATRIXJ1:func.7">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_2_1_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:74_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="matrixj1.html#K7" title="MATRIXJ1:func.7">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_2_1_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E3:74_2_1_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:74_2_1"/>
<font color="Maroon" title="c2">S</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c3">n</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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