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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being  <a href="matrixj1.html#NM1" title="MATRIXJ1:NM.1">FinSequence_of_Matrix</a> of <font color="Maroon" title="c1">D</font> holds   <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Olive" title="b3">F2</font></span>)</span>,<font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b1">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Olive" title="b3">F2</font></span>)</span>,<font color="Olive" title="b1">d</font></span><br/><b>let </b><font color="Maroon" title="c2">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being  <a href="matrixj1.html#NM1" title="MATRIXJ1:NM.1">FinSequence_of_Matrix</a> of <font color="Maroon" title="c1">D</font> holds   <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Olive" title="b2">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Olive" title="b1">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Olive" title="b2">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span><br/><b>let </b><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</font> be   <a href="matrixj1.html#NM1" title="MATRIXJ1:NM.1">FinSequence_of_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">  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span><br/>







<b>set </b><font color="Maroon" title="c5">F12</font> = <font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font>;<br/>
<b>set </b><font color="Maroon" title="c6">D1</font> =  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font>;<br/>
<b>set </b><font color="Maroon" title="c7">D2</font> =  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c2">d</font>;<br/>
<b>set </b><font color="Maroon" title="c8">D12</font> =  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font>;<br/>
<b>set </b><font color="Maroon" title="c9">DF</font> = <span class="p1"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font>;<br/>
<b>set </b><font color="Maroon" title="c10">DF2</font> =  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font>;<br/>
<b>set </b><font color="Maroon" title="c11">LF1</font> =  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font>;<br/>
<b>set </b><font color="Maroon" title="c12">WF1</font> =  <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font>;<br/>
<b>set </b><font color="Maroon" title="c13">LF2</font> =  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">F2</font>;<br/>
<b>set </b><font color="Maroon" title="c14">WF2</font> =  <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c4">F2</font>;<br/>
<b>set </b><font color="Maroon" title="c15">LF</font> =  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c16">WF</font> =  <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>;<br/>
<a NAME="E1:53"/><i><font color="Green" title="E42">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a></i>;<br/>
<a NAME="E2:53"/>
(  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">F2</font></span>)</span> &amp;  <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c4">F2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixj1.html#T14" target="_self" title="MATRIXJ1:th.14">Th14</a>, <a class="ref" href="matrixj1.html#T18" target="_self" title="MATRIXJ1:th.18">Th18</a></i>;<br/>
<a NAME="E3:53"/><b>then </b><i><font color="Green" title="E43">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</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="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">F2</font></span>)</span></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</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="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c4">F2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/>
<a NAME="E4:53"/><i><font color="Green" title="E44">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</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"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a></i>;<br/>
<a NAME="E5:53"/>
(  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">F2</font></span>)</span> &amp;  <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c4">F2</font></span>)</span> &amp;  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p1"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> &amp;  <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p1"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="matrixj1.html#T14" target="_self" title="MATRIXJ1:th.14">Th14</a>, <a class="ref" href="matrixj1.html#T15" target="_self" title="MATRIXJ1:th.15">Th15</a>, <a class="ref" href="matrixj1.html#T18" target="_self" title="MATRIXJ1:th.18">Th18</a>, <a class="ref" href="matrixj1.html#T19" target="_self" title="MATRIXJ1:th.19">Th19</a></i>;<br/>
<a NAME="E6:53"/><b>then </b><i><font color="Green" title="E45">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</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> <span class="p2">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</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="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">F2</font></span>)</span></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</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="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c4">F2</font></span>)</span></span>)</span> &amp;  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></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"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span> &amp;  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E4:53"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="rvsum_1.html#T103" title="RVSUM_1:th.103">RVSUM_1:103</a>, <a class="ref" href="rvsum_1.html#T106" title="RVSUM_1:th.106">RVSUM_1:106</a></i>;<br/>
<a NAME="E7:53"/><i><font color="Green" title="E46">A5</font></i>: 
(  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</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> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span> &amp;  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a></i>;<br/>
<i><font color="Green" title="E47">A6</font></i>:  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:53"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="#E4:53"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="#E6:53"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E9:53"/><i><font color="Green" title="E48">A7</font></i>: 
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span>,<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font>
 
<b>by </b><i><a class="ref" href="matrixj1.html#T34" target="_self" title="MATRIXJ1:th.34">Th34</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">i</font>, <font color="Maroon" title="c18">j</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font> )</span><br/><b>assume </b><a NAME="E1:53_2"/><i><font color="Green" title="E49">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</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"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font></span><br/><a NAME="E2:53_2"/><i><font color="Green" title="E50">A9</font></i>: 
( <font color="Maroon" title="c17">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"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:53_2"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:53_2"/><b>then </b><i><font color="Green" title="E51">A10</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_2_1_1"/>
( ( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font></span>)</span> ) or ( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font></span>)</span> ) or ( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font></span>)</span> ) or ( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_1"><b>suppose </b></a><a NAME="E1:53_2_1_1_1"/>
( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</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"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font></span><br/><div class="add"><a NAME="E2:53_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c17">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"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E7:53"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="#E2:53_2"><i><font color="Green" title="E50">A9</font></i></a>, <a class="txt" href="#E3:53_2"><i><font color="Green" title="E51">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="E3:53_2_1_1_1"/><b>then </b><i><font color="Green" title="E52">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font> = 
<span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font>
<b>by </b><i><a class="ref" href="matrixj1.html#T26" target="_self" title="MATRIXJ1:th.26">Th26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font>
<b>by </b><i><a class="ref" href="matrixj1.html#T26" target="_self" title="MATRIXJ1:th.26">Th26</a>, <a class="txt" href="#E3:53_2_1_1_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="txt" href="#E9:53"><i><font color="Green" title="E48">A7</font></i></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="53_2_1_1_2"><b>suppose </b></a><a NAME="E1:53_2_1_1_2"/><i><font color="Green" title="E52">A12</font></i>: 
( ( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font></span>)</span> ) or ( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</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"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font> = 
<font color="Maroon" title="c2">d</font>
<b>by </b><i><a class="ref" href="matrixj1.html#T29" target="_self" title="MATRIXJ1:th.29">Th29</a>, <a class="txt" href="#E1:53_2"><i><font color="Green" title="E49">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font>
<b>by </b><i><a class="txt" href="#E1:53_2_1_1_2"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="#E4:53"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="#E6:53"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="matrixj1.html#T29" target="_self" title="MATRIXJ1:th.29">Th29</a>, <a class="txt" href="#E8:53"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="#E1:53_2"><i><font color="Green" title="E49">A8</font></i></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="53_2_1_1_3"><b>suppose </b></a><a NAME="E1:53_2_1_1_3"/><i><font color="Green" title="E52">A13</font></i>: 
( <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</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"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c19">ii</font> = <font color="Maroon" title="c17">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span></span>)</span>, <font color="Maroon" title="c20">jj</font> = <font color="Maroon" title="c18">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font></span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/><a NAME="E3:53_2_1_1_3"/><i><font color="Green" title="E53">A14</font></i>: 
( <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ii</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">F1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">jj</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c3">F1</font></span>)</span></span>)</span> )
 ;<br/><a NAME="E4:53_2_1_1_3"/>
( <font color="Maroon" title="c19">ii</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c20">jj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="#E1:53_2_1_1_3"><i><font color="Green" title="E52">A13</font></i></a></i>;<br/><a NAME="E5:53_2_1_1_3"/><b>then </b>
( <font color="Maroon" title="c19">ii</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c20">jj</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/><a NAME="E6:53_2_1_1_3"/><b>then </b><i><font color="Green" title="E54">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">ii</font>,<font color="Maroon" title="c20">jj</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c2">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:53_2"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="matrixj1.html#T27" target="_self" title="MATRIXJ1:th.27">Th27</a>, <a class="txt" href="#E3:53_2_1_1_3"><i><font color="Green" title="E53">A14</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font> = 
<span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">ii</font>,<font color="Maroon" title="c20">jj</font>
<b>by </b><i><a class="txt" href="#E3:53_2_1_1_3"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="matrixj1.html#T28" target="_self" title="MATRIXJ1:th.28">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font>
<b>by </b><i><a class="txt" href="#E4:53"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="#E6:53"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="#E3:53_2_1_1_3"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="#E6:53_2_1_1_3"><i><font color="Green" title="E54">A15</font></i></a>, <a class="ref" href="matrixj1.html#T28" target="_self" title="MATRIXJ1:th.28">Th28</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="E5:53_2"/>
<span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</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="E11:53"/>
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c2">d</font></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="txt" href="#E3:53"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="#E4:53"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="#E6:53"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</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>
