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<b>let </b><font color="Maroon" title="c1">n</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">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">M</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">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="Olive" title="b5">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> holds <br/> <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><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>; <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">M</font> being  <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/> for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><br/> for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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="Olive" title="b4">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> holds <br/> <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">f</font></span>)</span>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font></span><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>; <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">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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="Olive" title="b3">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> holds <br/> <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b1">f</font></span>)</span>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</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">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="Olive" title="b2">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> holds <br/> <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><br/><b>let </b><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">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"> ( <font color="Maroon" title="c5">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="c6">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> implies  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> )</span><br/>











<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c5">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="c6">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">I</font> = <font color="Maroon" title="c5">i</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="E3:17"/><i><font color="Green" title="E18">A2</font></i>: 
(  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K9" title="MATRIX_2:func.9">Deleting</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">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="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K7" title="MATRIX_2:func.7">DelCol</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="c5">i</font></span>)</span>,<font color="Maroon" title="c6">j</font> &amp;  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K9" title="MATRIX_2:func.9">Deleting</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> &amp;  <a href="matrix_2.html#K9" title="MATRIX_2:func.9">Deleting</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K7" title="MATRIX_2:func.7">DelCol</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c6">j</font> )
 
<b>by </b><i><a class="txt" href="#E1:17"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="laplace.html#D1" target="_self" title="LAPLACE:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">f'</font> = <font color="Maroon" title="c4">f</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <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> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>suppose </b></a><a NAME="E1:17_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><br/><div class="add"><a NAME="E2:17_1_1"/><b>then </b>
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">f'</font>
 <b>by </b><i><a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1"/>
 <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="txt" href="#E3:17"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="comput_1.html#T6" title="COMPUT_1:th.6">COMPUT_1:6</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="17_1_2"><b>suppose </b></a><a NAME="E1:17_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><br/><div class="add"><b>hence </b><a NAME="E2:17_1_2"/>
 <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
 <b>by </b><i><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><b>end;</b></div>

</div>
