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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c3">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c4">V1</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c3">K</font>;<br/><b>let </b><font color="Maroon" title="c5">M</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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font>;<br/>









<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E39">A1</font></i>: 
( <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>  &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>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">d</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c3">K</font>;<br/>

<b>assume </b><a NAME="E2:50"/><i><font color="Green" title="E40">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</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> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">d</font> holds <br/><font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">M</font>,<font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) )
 ;<br/>

<b>let </b><font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c4">V1</font>;<br/>

<b>assume </b><a NAME="E3:50"/><i><font color="Green" title="E41">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">i</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">c</font> holds <br/><font color="Maroon" title="c9">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span> ) )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> $1,$2</span>)</span></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $2</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">n1</font> = <font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c2">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/>
<b>consider </b><font color="Maroon" title="c12">M1</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c10">n1</font>,<font color="Maroon" title="c11">m1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V1</font><b> such that </b><br/><a NAME="E6:50"/><i><font color="Green" title="E42">A4</font></i>: 
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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">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> <font color="Maroon" title="c12">M1</font> holds <br/><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)
 <b>from </b><i><a class="ref" href="matrix_1.html#S1" title="MATRIX_1:sch.1">MATRIX_1:sch 1</a>();<br/></i>
<a NAME="E7:50"/><i><font color="Green" title="E43">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M1</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="matrlin.html#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>
<a NAME="E8:50"/><i><font color="Green" title="E44">A6</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c12">M1</font> <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="matrlin.html#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>
<a NAME="E9:50"/><i><font color="Green" title="E45">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="matrlin.html#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>
<a NAME="E10:50"/><i><font color="Green" title="E46">A8</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">c</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="matrlin.html#E3:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E11:50"/><b>then </b><i><font color="Green" title="E47">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c9">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">p</font>
 
<b>by </b><i><a class="ref" href="matrlin.html#T16" target="_self" title="MATRLIN:th.16">Th16</a></i>;<br/>
<b>then </b><i><font color="Green" title="E48">A10</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c9">c</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M1</font>
<b>by </b><i><a class="txt" href="matrlin.html#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>
;<br/>
<div><i><font color="Green" title="E49">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:50_2"/><i><font color="Green" title="E50">A12</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c14">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c4">V1</font><b> such that </b><br/><a NAME="E3:50_2"/><i><font color="Green" title="E51">A13</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F</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="c8">b</font> &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">F</font> holds <br/><font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E4:50_2"/><i><font color="Green" title="E52">A14</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:50_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:50_2"/><i><font color="Green" title="E53">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">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="c12">M1</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="matrlin.html#E3:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="matrlin.html#E3:50_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/><a NAME="E6:50_2"/><i><font color="Green" title="E54">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">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="c5">M</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="matrlin.html#E3:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="matrlin.html#E3:50_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/><a NAME="E7:50_2"/><i><font color="Green" title="E55">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> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c13">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E5:50_2"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><i><font color="Green" title="E56">A18</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E6:50_2"><i><font color="Green" title="E54">A16</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="txt" href="matrlin.html#E3:50_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E9:50_2"/>
<font color="Maroon" title="c13">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:50_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:50_2"/><b>then </b>
<font color="Maroon" title="c13">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M1</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>;<br/><a NAME="E11:50_2"/><b>then </b><i><font color="Green" title="E57">A19</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">M1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E58">A20</font></i>: <font color="Maroon" title="c12">M1</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c13">i</font> = 
<font color="Maroon" title="c12">M1</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="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c13">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E11:50_2"><i><font color="Green" title="E57">A19</font></i></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></i>
;<br/><a NAME="E13:50_2"/><i><font color="Green" title="E59">X</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c13">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E12:50_2"><i><font color="Green" title="E58">A20</font></i></a>, <a class="txt" href="matrlin.html#E7:50_2"><i><font color="Green" title="E55">A17</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><i><font color="Green" title="E60">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:50_2_3"/><i><font color="Green" title="E61">A22</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c13">i</font></span>)</span>
 ;<br/><a NAME="E2:50_2_3"/><b>then </b><i><font color="Green" title="E62">A23</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:50_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="matrlin.html#E13:50_2"><i><font color="Green" title="E59">X</font></i></a></i>;<br/><a NAME="E3:50_2_3"/><b>then </b><i><font color="Green" title="E63">A24</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E8:50_2"><i><font color="Green" title="E56">A18</font></i></a>, <a class="ref" href="matrlin.html#T16" target="_self" title="MATRLIN:th.16">Th16</a></i>;<br/><a NAME="E4:50_2_3"/><i><font color="Green" title="E64">A25</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E2:50_2_3"><i><font color="Green" title="E62">A23</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E5:50_2_3"/><i><font color="Green" title="E65">A26</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E6:50_2"><i><font color="Green" title="E54">A16</font></i></a>, <a class="txt" href="matrlin.html#E1:50_2_3"><i><font color="Green" title="E61">A22</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrlin.html#E13:50_2"><i><font color="Green" title="E59">X</font></i></a></i>;<br/><a NAME="E6:50_2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c15">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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">M1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c12">M1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="matrlin.html#E5:50_2"><i><font color="Green" title="E53">A15</font></i></a>, <a class="txt" href="matrlin.html#E11:50_2"><i><font color="Green" title="E57">A19</font></i></a>, <a class="txt" href="matrlin.html#E1:50_2_3"><i><font color="Green" title="E61">A22</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a>, <a class="txt" href="matrlin.html#E13:50_2"><i><font color="Green" title="E59">X</font></i></a></i>;<br/><a NAME="E7:50_2_3"/><b>then </b><i><font color="Green" title="E66">A27</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c15">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> <font color="Maroon" title="c12">M1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></i>;<br/><i><font color="Green" title="E67">A28</font></i>: <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E3:50_2_3"><i><font color="Green" title="E63">A24</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c4">V1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E3:50_2_3"><i><font color="Green" title="E63">A24</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E4:50_2_3"><i><font color="Green" title="E64">A25</font></i></a>, <a class="txt" href="matrlin.html#E5:50_2_3"><i><font color="Green" title="E65">A26</font></i></a>, <a class="ref" href="vectsp_1.html#D24" title="VECTSP_1:def.24">VECTSP_1:def 24</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> = 
<font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E5:50_2"><i><font color="Green" title="E53">A15</font></i></a>, <a class="txt" href="matrlin.html#E12:50_2"><i><font color="Green" title="E58">A20</font></i></a>, <a class="txt" href="matrlin.html#E1:50_2_3"><i><font color="Green" title="E61">A22</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrlin.html#E13:50_2"><i><font color="Green" title="E59">X</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E6:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="matrlin.html#E7:50_2_3"><i><font color="Green" title="E66">A27</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E8:50_2_3"><i><font color="Green" title="E67">A28</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
<br/>.= 
<font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E3:50_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="matrlin.html#E1:50_2_3"><i><font color="Green" title="E61">A22</font></i></a>, <a class="txt" href="matrlin.html#E4:50_2"><i><font color="Green" title="E52">A14</font></i></a>, <a class="txt" href="matrlin.html#E13:50_2"><i><font color="Green" title="E59">X</font></i></a></i>
;<br/></div><b>end;</b></div><a NAME="E15:50_2"/><i><font color="Green" title="E61">A29</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">c</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E10:50"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="matrlin.html#E11:50"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="matrlin.html#E12:50"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="matrlin.html#E1:50_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E16:50_2"/><b>then </b><i><font color="Green" title="E62">A30</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E17:50_2"/><i><font color="Green" title="E63">A31</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E10:50"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="matrlin.html#E15:50_2"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E18:50_2"/><i><font color="Green" title="E64">A32</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c4">V1</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E12:50"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="matrlin.html#E1:50_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E19:50_2"/><i><font color="Green" title="E65">A33</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E8:50_2"><i><font color="Green" title="E56">A18</font></i></a>, <a class="ref" href="matrlin.html#T16" target="_self" title="MATRLIN:th.16">Th16</a></i>;<br/><a NAME="E20:50_2"/><b>then </b><i><font color="Green" title="E66">A34</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:50_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E21:50_2"/><i><font color="Green" title="E67">A35</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:50_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="matrlin.html#E19:50_2"><i><font color="Green" title="E65">A33</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E22:50_2"/><i><font color="Green" title="E68">A36</font></i>: 
for <font color="Olive" title="b1">j</font> being   <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>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">F</font> holds <br/><font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:50_2"><i><font color="Green" title="E51">A13</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c9">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> = 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c4">V1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E18:50_2"><i><font color="Green" title="E64">A32</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E16:50_2"><i><font color="Green" title="E62">A30</font></i></a>, <a class="txt" href="matrlin.html#E17:50_2"><i><font color="Green" title="E63">A31</font></i></a>, <a class="ref" href="vectsp_1.html#D24" title="VECTSP_1:def.24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E3:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="matrlin.html#E15:50_2"><i><font color="Green" title="E61">A29</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c14">F</font>
<b>by </b><i><a class="txt" href="matrlin.html#E20:50_2"><i><font color="Green" title="E66">A34</font></i></a>, <a class="txt" href="matrlin.html#E21:50_2"><i><font color="Green" title="E67">A35</font></i></a>, <a class="txt" href="matrlin.html#E22:50_2"><i><font color="Green" title="E68">A36</font></i></a>, <a class="ref" href="vectsp_3.html#T10" title="VECTSP_3:th.10">VECTSP_3:10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E7:50_2"><i><font color="Green" title="E55">A17</font></i></a>, <a class="txt" href="matrlin.html#E12:50_2"><i><font color="Green" title="E58">A20</font></i></a>, <a class="txt" href="matrlin.html#E14:50_2"><i><font color="Green" title="E60">A21</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:50_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span> <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="matrlin.html#E1:50_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E14:50"/><i><font color="Green" title="E50">A37</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="matrlin.html#E3:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E15:50"/><b>then </b><i><font color="Green" title="E51">A38</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">d</font>
 
<b>by </b><i><a class="ref" href="matrlin.html#T16" target="_self" title="MATRLIN:th.16">Th16</a></i>;<br/>
<a NAME="E16:50"/><i><font color="Green" title="E52">A39</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E14:50"><i><font color="Green" title="E50">A37</font></i></a>, <a class="ref" href="matrlin.html#T16" target="_self" title="MATRLIN:th.16">Th16</a></i>;<br/>
<i><font color="Green" title="E53">A40</font></i>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c12">M1</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>
;<br/>
<i><font color="Green" title="E54">A41</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">b</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="txt" href="matrlin.html#E16:50"><i><font color="Green" title="E52">A39</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="matrlin.html#E3:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="matrlin.html#E17:50"><i><font color="Green" title="E53">A40</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>
;<br/>
<a NAME="E19:50"/><i><font color="Green" title="E55">Y</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E18:50"><i><font color="Green" title="E54">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><i><font color="Green" title="E56">A42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:50_5"/><i><font color="Green" title="E57">A43</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">b</font></span>)</span>
 ;<br/><a NAME="E2:50_5"/><i><font color="Green" title="E58">A44</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">H</font> =  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c6">p</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c3">K</font> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">H</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c15">G</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c4">V1</font><b> such that </b><br/><a NAME="E5:50_5"/><i><font color="Green" title="E59">A45</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">G</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">p</font> &amp; ( for <font color="Olive" title="b1">i</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">G</font> holds <br/><font color="Maroon" title="c15">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E6:50_5"/><i><font color="Green" title="E60">A46</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E5:50_5"><i><font color="Green" title="E59">A45</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:50_5"/><i><font color="Green" title="E61">A48</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">G</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E5:50_5"><i><font color="Green" title="E59">A45</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><i><font color="Green" title="E62">A49</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"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c13">j</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <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"><font color="Maroon" title="c12">M1</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> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">G</font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="matrlin.html#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="matrlin.html#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="matrlin.html#E8:50"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="matrlin.html#E5:50_5"><i><font color="Green" title="E59">A45</font></i></a>, <a class="ref" href="matrix_2.html#T15" title="MATRIX_2:th.15">MATRIX_2:15</a></i>
;<br/><a NAME="E9:50_5"/><i><font color="Green" title="E63">A50</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="matrlin.html#E9:50"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/><a NAME="E10:50_5"/><b>then </b><i><font color="Green" title="E64">A51</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T86" title="FINSEQ_2:th.86">FINSEQ_2:86</a></i>;<br/><a NAME="E11:50_5"/><i><font color="Green" title="E65">X</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c13">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="matrlin.html#E8:50_5"><i><font color="Green" title="E62">A49</font></i></a></i>;<br/><div><i><font color="Green" title="E66">A52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:50_5_2"/><i><font color="Green" title="E67">A53</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c13">j</font></span>)</span>
 ;<br/><a NAME="E2:50_5_2"/><b>then </b><i><font color="Green" title="E68">A54</font></i>: 
<font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E8:50_5"><i><font color="Green" title="E62">A49</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrlin.html#E11:50_5"><i><font color="Green" title="E65">X</font></i></a></i>;<br/><a NAME="E3:50_5_2"/><i><font color="Green" title="E69">A55</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">G</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:50_5_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">G</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="matrlin.html#E7:50"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="matrlin.html#E5:50_5"><i><font color="Green" title="E59">A45</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E5:50_5_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c13">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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">M1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c12">M1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="matrlin.html#E17:50"><i><font color="Green" title="E53">A40</font></i></a>, <a class="txt" href="matrlin.html#E1:50_5"><i><font color="Green" title="E57">A43</font></i></a>, <a class="txt" href="matrlin.html#E1:50_5_2"><i><font color="Green" title="E67">A53</font></i></a>, <a class="txt" href="matrlin.html#E3:50_5_2"><i><font color="Green" title="E69">A55</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a>, <a class="txt" href="matrlin.html#E11:50_5"><i><font color="Green" title="E65">X</font></i></a>, <a class="txt" href="matrlin.html#E19:50"><i><font color="Green" title="E55">Y</font></i></a></i>;<br/><a NAME="E6:50_5_2"/><b>then </b><i><font color="Green" title="E70">A56</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c12">M1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></i>;<br/><a NAME="E7:50_5_2"/><i><font color="Green" title="E71">A57</font></i>: 
<font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c6">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E5:50_5"><i><font color="Green" title="E59">A45</font></i></a>, <a class="txt" href="matrlin.html#E9:50_5"><i><font color="Green" title="E63">A50</font></i></a>, <a class="txt" href="matrlin.html#E1:50_5_2"><i><font color="Green" title="E67">A53</font></i></a>, <a class="ref" href="finseq_2.html#T86" title="FINSEQ_2:th.86">FINSEQ_2:86</a>, <a class="txt" href="matrlin.html#E11:50_5"><i><font color="Green" title="E65">X</font></i></a></i>;<br/><a NAME="E8:50_5_2"/><b>then </b><i><font color="Green" title="E72">A58</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">H</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:50_5_2"/><i><font color="Green" title="E73">A59</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E7:50_5_2"><i><font color="Green" title="E71">A57</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:50_5_2"/><i><font color="Green" title="E74">A60</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green" title="E75">A61</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c13">j</font></span>)</span> = 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c13">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E7:50_5"><i><font color="Green" title="E61">A48</font></i></a>, <a class="txt" href="matrlin.html#E1:50_5_2"><i><font color="Green" title="E67">A53</font></i></a>, <a class="txt" href="matrlin.html#E3:50_5_2"><i><font color="Green" title="E69">A55</font></i></a>, <a class="txt" href="matrlin.html#E11:50_5"><i><font color="Green" title="E65">X</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="matrlin.html#E9:50"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="matrlin.html#E5:50_5"><i><font color="Green" title="E59">A45</font></i></a>, <a class="txt" href="matrlin.html#E1:50_5_2"><i><font color="Green" title="E67">A53</font></i></a>, <a class="txt" href="matrlin.html#E10:50_5_2"><i><font color="Green" title="E74">A60</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrlin.html#E11:50_5"><i><font color="Green" title="E65">X</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c13">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E9:50_5_2"><i><font color="Green" title="E73">A59</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c14">H</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E8:50_5_2"><i><font color="Green" title="E72">A58</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><b>thus </b><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"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c13">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E2:50_5_2"><i><font color="Green" title="E68">A54</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c13">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E6:50_5_2"><i><font color="Green" title="E70">A56</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"><font color="Maroon" title="c14">H</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E6:50"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="matrlin.html#E6:50_5_2"><i><font color="Green" title="E70">A56</font></i></a>, <a class="txt" href="matrlin.html#E11:50_5_2"><i><font color="Green" title="E75">A61</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E5:50_5"><i><font color="Green" title="E59">A45</font></i></a>, <a class="txt" href="matrlin.html#E1:50_5_2"><i><font color="Green" title="E67">A53</font></i></a>, <a class="txt" href="matrlin.html#E6:50_5"><i><font color="Green" title="E60">A46</font></i></a>, <a class="txt" href="matrlin.html#E11:50_5"><i><font color="Green" title="E65">X</font></i></a></i>
;<br/></div><b>end;</b></div><a NAME="E13:50_5"/><i><font color="Green" title="E67">A62</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:50_5"><i><font color="Green" title="E57">A43</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="matrlin.html#E19:50"><i><font color="Green" title="E55">Y</font></i></a></i>;<br/><a NAME="E14:50_5"/><i><font color="Green" title="E68">A63</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:50_5"><i><font color="Green" title="E57">A43</font></i></a></i>;<br/><a NAME="E15:50_5"/><b>then </b><i><font color="Green" title="E69">A64</font></i>: 
<font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E15:50"><i><font color="Green" title="E51">A38</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E16:50_5"/><i><font color="Green" title="E70">A65</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E16:50"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="matrlin.html#E14:50_5"><i><font color="Green" title="E68">A63</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E17:50_5"/><i><font color="Green" title="E71">A66</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E14:50"><i><font color="Green" title="E50">A37</font></i></a>, <a class="txt" href="matrlin.html#E14:50_5"><i><font color="Green" title="E68">A63</font></i></a>, <a class="ref" href="matrlin.html#T16" target="_self" title="MATRLIN:th.16">Th16</a></i>;<br/><a NAME="E18:50_5"/><i><font color="Green" title="E72">A67</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">G</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E5:50_5"><i><font color="Green" title="E59">A45</font></i></a>, <a class="txt" href="matrlin.html#E10:50_5"><i><font color="Green" title="E64">A51</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E19:50_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>;<br/><a NAME="E20:50_5"/><b>then </b><i><font color="Green" title="E73">A68</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>then </b><i><font color="Green" title="E74">A69</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c13">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E13:50_5"><i><font color="Green" title="E67">A62</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c13">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E13:50_5"><i><font color="Green" title="E67">A62</font></i></a>, <a class="txt" href="matrlin.html#E20:50_5"><i><font color="Green" title="E73">A68</font></i></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> = 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c4">V1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E14:50_5"><i><font color="Green" title="E68">A63</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E15:50_5"><i><font color="Green" title="E69">A64</font></i></a>, <a class="txt" href="matrlin.html#E16:50_5"><i><font color="Green" title="E70">A65</font></i></a>, <a class="ref" href="vectsp_1.html#D24" title="VECTSP_1:def.24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c14">H</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E2:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="matrlin.html#E17:50_5"><i><font color="Green" title="E71">A66</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c15">G</font>
<b>by </b><i><a class="txt" href="matrlin.html#E5:50_5"><i><font color="Green" title="E59">A45</font></i></a>, <a class="txt" href="matrlin.html#E10:50_5"><i><font color="Green" title="E64">A51</font></i></a>, <a class="txt" href="matrlin.html#E18:50_5"><i><font color="Green" title="E72">A67</font></i></a>, <a class="ref" href="matrlin.html#T14" target="_self" title="MATRLIN:th.14">Th14</a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E8:50_5"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="matrlin.html#E12:50_5"><i><font color="Green" title="E66">A52</font></i></a>, <a class="txt" href="matrlin.html#E21:50_5"><i><font color="Green" title="E74">A69</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E13:50_5"><i><font color="Green" title="E67">A62</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:50_5"><i><font color="Green" title="E57">A43</font></i></a>, <a class="txt" href="matrlin.html#E2:50_5"><i><font color="Green" title="E58">A44</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a>, <a class="txt" href="matrlin.html#E19:50"><i><font color="Green" title="E55">Y</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>thus </b> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c9">c</font></span>)</span> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E12:50"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="matrlin.html#E13:50"><i><font color="Green" title="E49">A11</font></i></a>, <a class="ref" href="newton.html#T3" title="NEWTON:th.3">NEWTON:3</a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#T37" target="_self" title="MATRLIN:th.37">Th37</a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">b</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E18:50"><i><font color="Green" title="E54">A41</font></i></a>, <a class="txt" href="matrlin.html#E20:50"><i><font color="Green" title="E56">A42</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>
;<br/>


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