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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">c</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"> for <font color="Olive" title="b1">D</font> being 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> <br/> for <font color="Olive" title="b2">pD</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b1">D</font><br/> for <font color="Olive" title="b3">A</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">D</font><br/> for <font color="Olive" title="b4">A'</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">D</font>  st <font color="Olive" title="b4">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  &amp; ( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) holds <br/> <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Olive" title="b3">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b2">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Olive" title="b4">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b2">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span><br/><b>let </b><font color="Maroon" title="c4">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">pD</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">D</font><br/> for <font color="Olive" title="b2">A</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c4">D</font><br/> for <font color="Olive" title="b3">A'</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">D</font>  st <font color="Olive" title="b3">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  &amp; ( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) holds <br/> <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Olive" title="b2">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Olive" title="b3">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span><br/><b>let </b><font color="Maroon" title="c5">pD</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">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">A</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c4">D</font><br/> for <font color="Olive" title="b2">A'</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">D</font>  st <font color="Olive" title="b2">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  &amp; ( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) holds <br/> <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Olive" title="b2">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span><br/><b>let </b><font color="Maroon" title="c6">A</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c4">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">A'</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">D</font>  st <font color="Olive" title="b1">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  &amp; ( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) holds <br/> <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span><br/><b>let </b><font color="Maroon" title="c7">A'</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  &amp; ( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) implies  <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  )</span><br/>













<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c7">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span><br/>

<b>set </b><font color="Maroon" title="c8">RL</font> =  <a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font>;<br/>
<b>set </b><font color="Maroon" title="c9">RC</font> =  <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">pD</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c1">n</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>  ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">pD</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="c6">A</font> &amp; <font color="Maroon" title="c1">n</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>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span><br/><div class="add"><a NAME="E2:23_1_1_1"/><b>then </b>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <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="c6">A</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><b>then </b> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">A</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">A'</font>
<b>by </b><i><a class="txt" href="laplace.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
;<br/><a NAME="E4:23_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><a NAME="E5:23_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><a NAME="E6:23_1_1_1"/><b>then </b>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>;<br/><a NAME="E7:23_1_1_1"/><b>then </b>
(  <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"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="laplace.html#E1:23_1_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/><a NAME="E8:23_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><b>hence </b><a NAME="E9:23_1_1_1"/>
 <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
 ; <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="23_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2"/><i><font color="Green" title="E21">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">pD</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c1">n</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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span><br/><div class="add"><a NAME="E2:23_1_1_2"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c2">m</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>  )
 <b>by </b><i><a class="txt" href="laplace.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/><a NAME="E3:23_1_1_2"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">A'</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="c6">A</font>
 <b>by </b><i><a class="txt" href="laplace.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/><b>thus </b> <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> = 
<font color="Maroon" title="c6">A</font>
<b>by </b><i><a class="txt" href="laplace.html#E1:23_1_1_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="laplace.html#D2" target="_self" title="LAPLACE:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
<b>by </b><i><a class="txt" href="laplace.html#E1:23_1_1_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="laplace.html#E2:23_1_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="matrix_2.html#T15" title="MATRIX_2:th.15">MATRIX_2:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
<b>by </b><i><a class="txt" href="laplace.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="laplace.html#E1:23_1_1_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="laplace.html#E3:23_1_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</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="23_1_1_3"><b>suppose </b></a><a NAME="E1:23_1_1_3"/><i><font color="Green" title="E21">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">pD</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="c6">A</font> &amp; <font color="Maroon" title="c1">n</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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span><br/><div class="add"><a NAME="E2:23_1_1_3"/><b>then </b><i><font color="Green" title="E22">A8</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">m</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>  )
 <b>by </b><i><a class="txt" href="laplace.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/><a NAME="E3:23_1_1_3"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/><a NAME="E4:23_1_1_3"/><b>then </b>
(  <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"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="txt" href="laplace.html#E1:23_1_1_3"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">RL'</font> = <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  as    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c4">D</font> <b>by </b><i><a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c12">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="c11">i</font>,<font color="Maroon" title="c12">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="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> implies <span class="p1">(<span class="default"><a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">RL'</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )</span><br/><b>assume </b><a NAME="E1:23_1_1_3_1"/><i><font color="Green" title="E24">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">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="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</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="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">RL'</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><br/><b>reconsider </b><font color="Maroon" title="c13">I</font> = <font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c14">J</font> = <font color="Maroon" title="c12">j</font> 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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:23_1_1_3_1"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c10">RL'</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/><a NAME="E4:23_1_1_3_1"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c11">i</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="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span>
 <b>by </b><i><a class="txt" href="laplace.html#E1:23_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/><a NAME="E5:23_1_1_3_1"/><i><font color="Green" title="E26">A12</font></i>: 
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">A</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c7">A'</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_3_1_1"/>
( <font color="Maroon" title="c14">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> or <font color="Maroon" title="c14">J</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_3_1_1_1"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c14">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</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"><a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">RL'</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font> = 
<font color="Maroon" title="c5">pD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">I</font>
<b>by </b><i><a class="txt" href="laplace.html#E1:23_1_1_3"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="laplace.html#E1:23_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="laplace.html#E5:23_1_1_3_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="laplace.html#D2" target="_self" title="LAPLACE:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">J</font>,<font color="Maroon" title="c13">I</font>
<b>by </b><i><a class="txt" href="laplace.html#E1:23_1_1_3"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="laplace.html#E2:23_1_1_3"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="laplace.html#E3:23_1_1_3"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="laplace.html#E4:23_1_1_3_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="laplace.html#E5:23_1_1_3_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="laplace.html#E1:23_1_1_3_1_1_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c10">RL'</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="txt" href="laplace.html#E4:23_1_1_3_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_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="23_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_3_1_1_2"/><i><font color="Green" title="E27">A14</font></i>: 
<font color="Maroon" title="c14">J</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</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"><a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">RL'</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font> = 
<font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">I</font>,<font color="Maroon" title="c14">J</font>
<b>by </b><i><a class="txt" href="laplace.html#E1:23_1_1_3"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="laplace.html#E1:23_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="laplace.html#E5:23_1_1_3_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="laplace.html#D2" target="_self" title="LAPLACE:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c7">A'</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c11">i</font>
<b>by </b><i><a class="txt" href="laplace.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="laplace.html#E1:23_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="laplace.html#E5:23_1_1_3_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">J</font>,<font color="Maroon" title="c13">I</font>
<b>by </b><i><a class="txt" href="laplace.html#E1:23_1_1_3"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="laplace.html#E2:23_1_1_3"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="laplace.html#E3:23_1_1_3"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="laplace.html#E4:23_1_1_3_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="laplace.html#E5:23_1_1_3_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="laplace.html#E1:23_1_1_3_1_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c10">RL'</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="txt" href="laplace.html#E4:23_1_1_3_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_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="E7:23_1_1_3"/>
 <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
 <b>by </b><i><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</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><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:23"/>
 <a href="laplace.html#K3" title="LAPLACE:func.3">ReplaceCol</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">pD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
 ; <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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