<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>;<br/><b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>









<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">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> <font color="Maroon" title="c1">n</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c7">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E2:15"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c6">k</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"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c7">m</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"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">I</font> = <font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c9">J</font> = <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c10">K</font> = <font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c11">U</font> = <font color="Maroon" title="c7">m</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="E4:15"/>
<font color="Maroon" title="c1">n</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="laplace.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E5:15"/><b>then </b>
<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> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">n'</font> = <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<a NAME="E7:15"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n'</font>
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">DL</font> =  <a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>;<br/>
<b>set </b><font color="Maroon" title="c14">DM</font> =  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>;<br/>
<a NAME="E8:15"/><i><font color="Green" title="E18">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font> <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#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E9:15"/><b>then </b><i><font color="Green" title="E19">A5</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n'</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</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="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E10:15"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n'</font>
 
<b>by </b><i><a class="txt" href="laplace.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="laplace.html#E8:15"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="laplace.html#T1" target="_self" title="LAPLACE:th.1">Th1</a></i>;<br/>
<a NAME="E11:15"/><b>then </b>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</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> <font color="Maroon" title="c12">n'</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</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> <font color="Maroon" title="c12">n'</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n'</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="laplace.html#E9:15"><i><font color="Green" title="E19">A5</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="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E12:15"/><b>then </b><i><font color="Green" title="E20">A6</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix_2.html#K9" title="MATRIX_2:func.9">Deleting</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c5">j</font> &amp;  <a href="matrix_2.html#K9" title="MATRIX_2:func.9">Deleting</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> &amp; <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span>,<font color="Maroon" title="c6">k</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">m</font> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="laplace.html#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="laplace.html#D1" target="_self" title="LAPLACE:def.1">Def1</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a>, <a class="ref" href="matrix_2.html#D6" title="MATRIX_2:def.6">MATRIX_2:def 6</a></i>;<br/>
<a NAME="E13:15"/>
<font color="Maroon" title="c12">n'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E14:15"/><b>then </b><i><font color="Green" title="E21">A7</font></i>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">n'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><font color="Maroon" title="c12">n'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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"><font color="Maroon" title="c12">n'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="laplace.html#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a>, <a class="ref" href="finseq_1.html#T81" title="FINSEQ_1:th.81">FINSEQ_1:81</a></i>;<br/>
<a NAME="E15:15"/><i><font color="Green" title="E22">A8</font></i>: 
( <font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">I</font> implies ( ( <font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> ) &amp; ( <font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">I</font>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">K</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="laplace.html#E8:15"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="laplace.html#E9:15"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="laplace.html#E14:15"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></i>;<br/>
<b>thus </b><a NAME="E3:15_1"/>
( <font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">J</font>
 ;<br/>

<a NAME="E2:15_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c10">K</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> <font color="Maroon" title="c3">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="laplace.html#E12:15"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="laplace.html#E2:15_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E3:15_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">U</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="laplace.html#E12:15"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="laplace.html#E14:15"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="laplace.html#E2:15_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="laplace.html#E1:15_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_1"/>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:15_1"/><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">J</font>
 ;<br/>

<a NAME="E5:15_1"/><i><font color="Green" title="E25">A12</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c10">K</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> <font color="Maroon" title="c3">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E12:15"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="laplace.html#E2:15_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E6:15_1"/><b>then </b>
( <font color="Maroon" title="c9">J</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c10">K</font></span>)</span> &amp; <font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n'</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="laplace.html#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:15_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="laplace.html#E12:15"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="laplace.html#E14:15"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="laplace.html#E2:15_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="laplace.html#E4:15_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="laplace.html#E5:15_1"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E8:15_1"/>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:15"/>
( <font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">I</font> implies ( ( <font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">U</font> ) &amp; ( <font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">I</font>
 ;<br/>

<a NAME="E2:15_2"/>
<font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n'</font>
 
<b>by </b><i><a class="txt" href="laplace.html#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:15_2"/><b>then </b><i><font color="Green" title="E24">A14</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="laplace.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="laplace.html#E8:15"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="laplace.html#E9:15"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="laplace.html#E14:15"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="laplace.html#E1:15_2"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></i>;<br/>
<b>thus </b><a NAME="E4:15_2"/>
( <font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">U</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2_1"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">J</font>
 ;<br/>

<a NAME="E2:15_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c10">K</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> <font color="Maroon" title="c3">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E12:15"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="laplace.html#E3:15_2"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E3:15_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">U</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">U</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="laplace.html#E12:15"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="laplace.html#E14:15"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="laplace.html#E3:15_2"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="laplace.html#E1:15_2_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:15_2_1"/>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">U</font>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:15_2"/><i><font color="Green" title="E25">A16</font></i>: 
<font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">J</font>
 ;<br/>

<a NAME="E6:15_2"/><i><font color="Green" title="E26">A17</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c10">K</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> <font color="Maroon" title="c3">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E12:15"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="laplace.html#E3:15_2"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E7:15_2"/><b>then </b>
( <font color="Maroon" title="c9">J</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c10">K</font></span>)</span> &amp; <font color="Maroon" title="c11">U</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n'</font> )
 
<b>by </b><i><a class="txt" href="laplace.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="laplace.html#E2:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="laplace.html#E7:15"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:15_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="laplace.html#E12:15"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="laplace.html#E14:15"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="laplace.html#E3:15_2"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="laplace.html#E5:15_2"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="laplace.html#E6:15_2"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E9:15_2"/>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">K</font>,<font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:15"/>
( ( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">m</font> ) &amp; ( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c7">m</font> ) &amp; ( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="laplace.html#E15:15"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/>


</div>
