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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">V1</font> being   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Olive" title="b1">K</font><br/>  for <font color="Olive" title="b3">M</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">d</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Olive" title="b1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">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="Olive" title="b5">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="b6">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">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="Olive" title="b5">d</font> holds <br/><font color="Olive" title="b5">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<font color="Olive" title="b4">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Olive" title="b3">M</font>,<font color="Olive" title="b6">j</font>)</span>)</span>)</span>)</span> ) holds <br/> for <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Olive" title="b2">V1</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">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="Olive" title="b7">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="b8">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b8">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="Olive" title="b7">c</font> holds <br/><font color="Olive" title="b7">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b3">M</font>,<font color="Olive" title="b8">i</font>)</span>)</span>,<font color="Olive" title="b6">b</font>)</span>)</span> ) holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Olive" title="b4">p</font>,<font color="Olive" title="b7">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Olive" title="b5">d</font>,<font color="Olive" title="b6">b</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V1</font> being   <a href="matrlin.html#V1" title="MATRLIN:attr.1">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/>  for <font color="Olive" title="b2">M</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">d</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c3">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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="Olive" title="b4">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="b5">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">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="Olive" title="b4">d</font> holds <br/><font color="Olive" title="b4">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<font color="Olive" title="b3">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b5">j</font>)</span>)</span>)</span>)</span> ) holds <br/> for <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Olive" title="b1">V1</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">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="Olive" title="b6">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="b7">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b7">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="Olive" title="b6">c</font> holds <br/><font color="Olive" title="b6">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b7">i</font>)</span>)</span>,<font color="Olive" title="b5">b</font>)</span>)</span> ) holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Olive" title="b3">p</font>,<font color="Olive" title="b6">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Olive" title="b4">d</font>,<font color="Olive" title="b5">b</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">V1</font> be   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c3">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">d</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c3">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">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="Olive" title="b3">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="b4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">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="Olive" title="b3">d</font> holds <br/><font color="Olive" title="b3">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<font color="Olive" title="b2">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b4">j</font>)</span>)</span>)</span>)</span> ) holds <br/> for <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c4">V1</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">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="Olive" title="b5">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="b6">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">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="Olive" title="b5">c</font> holds <br/><font color="Olive" title="b5">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b6">i</font>)</span>)</span>,<font color="Olive" title="b4">b</font>)</span>)</span> ) holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b5">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Olive" title="b3">d</font>,<font color="Olive" title="b4">b</font>)</span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">d</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c3">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b2">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="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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="Olive" title="b2">d</font> holds <br/><font color="Olive" title="b2">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<font color="Olive" title="b1">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="b3">j</font>)</span>)</span>)</span>)</span> ) holds <br/> for <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c4">V1</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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="Olive" title="b4">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="b5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">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="Olive" title="b4">c</font> holds <br/><font color="Olive" title="b4">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">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="b5">i</font>)</span>)</span>,<font color="Olive" title="b3">b</font>)</span>)</span> ) holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">b</font>)</span>)</span> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">d</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c3">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b2">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="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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="Olive" title="b2">d</font> holds <br/><font color="Olive" title="b2">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<font color="Olive" title="b1">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="b3">j</font>)</span>)</span>)</span>)</span> ) holds <br/> for <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c4">V1</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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="Olive" title="b4">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="b5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">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="Olive" title="b4">c</font> holds <br/><font color="Olive" title="b4">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">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="b5">i</font>)</span>)</span>,<font color="Olive" title="b3">b</font>)</span>)</span> ) holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">b</font>)</span>)</span></span><br/>

<a NAME="E3:45"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">n1</font> = <font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c7">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">d</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c3">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c9">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="c9">d</font> holds <br/><font color="Maroon" title="c9">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#K4" title="RLVECT_1:func.4">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="c8">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> ) implies  for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c4">V1</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b2">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="b3">i</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">c</font> holds <br/><font color="Olive" title="b2">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">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="b3">i</font>)</span>)</span>,<font color="Olive" title="b1">b</font>)</span>)</span> ) holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c8">p</font>,<font color="Olive" title="b2">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Olive" title="b1">b</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:45"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E6:45"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">and </span><br/><a NAME="E7:45"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
 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="c9">d</font> holds <br/><font color="Maroon" title="c9">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#K4" title="RLVECT_1:func.4">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="c8">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c4">V1</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b2">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="b3">i</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">c</font> holds <br/><font color="Olive" title="b2">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">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="b3">i</font>)</span>)</span>,<font color="Olive" title="b1">b</font>)</span>)</span> ) holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c8">p</font>,<font color="Olive" title="b2">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Olive" title="b1">b</font>)</span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">b</font>, <font color="Maroon" title="c11">c</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c4">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">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="c11">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="c11">c</font> holds <br/><font color="Maroon" title="c11">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#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">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="c10">b</font>)</span>)</span> ) implies  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c11">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E8:45"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">and </span><br/><a NAME="E9:45"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E10:45"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
 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="c11">c</font> holds <br/><font color="Maroon" title="c11">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#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">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="c10">b</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c11">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span><br/>

<span class="kw">deffunc </span><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>) <span class="kw">-&gt; </span>   <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="c8">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="c10">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $2</span>)</span>;<br/>
<span class="kw">consider </span><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="c6">n1</font>,<font color="Maroon" title="c7">m1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V1</font><span class="kw"> such that </span><br/><a NAME="E12:45"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
 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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="matrix_1.html#S1" title="MATRIX_1:sch.1">MATRIX_1:sch 1</a>();<br/></span>
<span class="lab"><font color="Green" title="E47">A11</font></span>:  <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D6" target="_self" title="MATRLIN:def.6">Def8</a></span>
;<br/>
<a NAME="E14:45"/><span class="lab"><font color="Green" title="E48">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">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="c10">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E15:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A13</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">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="c10">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T12" target="_self" title="MATRLIN:th.12">Th16</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E50">A14</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E13:45"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>
;<br/>
<a NAME="E17:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A15</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">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#K6" title="MATRLIN:func.6">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E18:45"/><span class="lab"><font color="Green" title="E52">A16</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<a NAME="E19:45"/><span class="lab"><font color="Green" title="E53">A17</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<a NAME="E20:45"/><span class="lab"><font color="Green" title="E54">A18</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">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="c9">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="matrlin.html#T12" target="_self" title="MATRLIN:th.12">Th16</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E55">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:45_3"/><span class="lab"><font color="Green" title="E56">A20</font></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="matrlin.html#K6" title="MATRLIN:func.6">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#K6" title="MATRLIN:func.6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c13">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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> )</span><br/><span class="kw">assume </span><a NAME="E2:45_3"/><span class="lab"><font color="Green" title="E57">A21</font></span>: 
<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#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">b</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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></span><br/><a NAME="E3:45_3"/><span class="lab"><font color="Green" title="E58">A22</font></span>: 
<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#K6" title="MATRLIN:func.6">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="txt" href="#E2:45_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:45_3"/><span class="lab"><font color="Green" title="E59">A23</font></span>: 
<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="c9">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="txt" href="#E2:45_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="matrlin.html#T12" target="_self" title="MATRLIN:th.12">Th16</a></span>;<br/><a NAME="E5:45_3"/><span class="lab"><font color="Green" title="E60">A24</font></span>: 
( <font color="Maroon" title="c9">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="c9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c10">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="c10">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:45"><span class="lab"><font color="Green" title="E54">A18</font></span></a>, <a class="txt" href="#E15:45"><span class="lab"><font color="Green" title="E49">A13</font></span></a>, <a class="txt" href="#E2:45_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E6:45_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D6" target="_self" title="MATRLIN:def.6">Def8</a></span>;<br/><a NAME="E7:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A25</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A26</font></span>: <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="pre_poly.html#K9" title="PRE_POLY:func.9">/.</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_3"><span class="lab"><font color="Green" title="E58">A22</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_3"><span class="lab"><font color="Green" title="E58">A22</font></span></a>, <a class="txt" href="#E7:45_3"><span class="lab"><font color="Green" title="E61">A25</font></span></a>, <a class="ref" href="matrix_2.html#T16" title="MATRIX_2:th.16">MATRIX_2:16</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">H</font> =  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<font color="Maroon" title="c8">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/><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <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="c10">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font></span>)</span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E11:45_3"/><span class="lab"><font color="Green" title="E63">A27</font></span>: 
(  <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="c8">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>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span><a NAME="E12:45_3"/><span class="lab"><font color="Green" title="E64">A28</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E13:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A29</font></span>: 
 <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="c8">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="c8">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></span>;<br/><a NAME="E14:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A30</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_3"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><a NAME="E15:45_3"/><span class="lab"><font color="Green" title="E67">A31</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_3"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><span class="lab"><font color="Green" title="E68">A32</font></span>:  <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E19:45"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="txt" href="#E18:45"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="txt" href="#E11:45_3"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="ref" href="matrix_2.html#T13" title="MATRIX_2:th.13">MATRIX_2:13</a></span>
;<br/><a NAME="E17:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A33</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E18:45_3"/><span class="lab"><font color="Green" title="E70">A34</font></span>: 
 <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="c8">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_3"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E71">A35</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )</span><br/><a NAME="E1:45_3_3"/><span class="lab"><font color="Green" title="E72">A36</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">assume </span><a NAME="E2:45_3_3"/><span class="lab"><font color="Green" title="E73">A37</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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></span><br/><a NAME="E3:45_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A38</font></span>: 
<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="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_3"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="txt" href="#E12:45_3"><span class="lab"><font color="Green" title="E64">A28</font></span></a>, <a class="txt" href="#E17:45_3"><span class="lab"><font color="Green" title="E69">A33</font></span></a>, <a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></span>;<br/><a NAME="E4:45_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A39</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E5:45_3_3"/><span class="lab"><font color="Green" title="E76">A40</font></span>: 
<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="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_3_3"><span class="lab"><font color="Green" title="E74">A38</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:45_3_3"/><span class="lab"><font color="Green" title="E77">A41</font></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> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A42</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c8">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="c8">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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:45_3"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E17:45_3"><span class="lab"><font color="Green" title="E69">A33</font></span></a>, <a class="txt" href="#E2:45_3_3"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<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="c8">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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E11:45_3"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="txt" href="#E17:45_3"><span class="lab"><font color="Green" title="E69">A33</font></span></a>, <a class="txt" href="#E2:45_3_3"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="txt" href="#E1:45_3_3"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<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="c8">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_3_3"><span class="lab"><font color="Green" title="E76">A40</font></span></a>, <a class="ref" href="funcop_1.html#T22" title="FUNCOP_1:th.22">FUNCOP_1:22</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_3_3"><span class="lab"><font color="Green" title="E75">A39</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><a NAME="E8:45_3_3"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E19:45"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="txt" href="#E11:45_3"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><a NAME="E9:45_3_3"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:45"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="txt" href="#E2:45_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="#E17:45_3"><span class="lab"><font color="Green" title="E69">A33</font></span></a>, <a class="txt" href="#E2:45_3_3"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="txt" href="#E6:45_3_3"><span class="lab"><font color="Green" title="E77">A41</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E10:45_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A43</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></span>;<br/><a NAME="E11:45_3_3"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45_3"><span class="lab"><font color="Green" title="E68">A32</font></span></a>, <a class="txt" href="#E17:45_3"><span class="lab"><font color="Green" title="E69">A33</font></span></a>, <a class="txt" href="#E2:45_3_3"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><span class="kw">hence </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45_3_3"><span class="lab"><font color="Green" title="E79">A43</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>
<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="c10">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:45"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E10:45_3_3"><span class="lab"><font color="Green" title="E79">A43</font></span></a>, <a class="txt" href="#E7:45_3_3"><span class="lab"><font color="Green" title="E78">A42</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_3"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="txt" href="#E18:45_3"><span class="lab"><font color="Green" title="E70">A34</font></span></a>, <a class="txt" href="#E17:45_3"><span class="lab"><font color="Green" title="E69">A33</font></span></a>, <a class="txt" href="#E2:45_3_3"><span class="lab"><font color="Green" title="E73">A37</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">thus </span><span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">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="c9">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="c10">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="funcop_1.html#T22" title="FUNCOP_1:th.22">FUNCOP_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c10">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_3"><span class="lab"><font color="Green" title="E60">A24</font></span></a>, <a class="ref" href="vectsp_1.html#D12" title="VECTSP_1:def.12">VECTSP_1:def 12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">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="c10">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E4:45_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a></span>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_3"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="txt" href="#E13:45_3"><span class="lab"><font color="Green" title="E65">A29</font></span></a>, <a class="txt" href="#E14:45_3"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="ref" href="matrlin.html#T10" target="_self" title="MATRLIN:th.10">Th14</a></span>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">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="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45_3"><span class="lab"><font color="Green" title="E68">A32</font></span></a>, <a class="txt" href="#E19:45_3"><span class="lab"><font color="Green" title="E71">A35</font></span></a>, <a class="txt" href="#E8:45_3"><span class="lab"><font color="Green" title="E62">A26</font></span></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_3"><span class="lab"><font color="Green" title="E58">A22</font></span></a>, <a class="ref" href="matrlin.html#D6" target="_self" title="MATRLIN:def.6">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="txt" href="#E2:45_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="#E1:45_3"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E22:45"/><span class="lab"><font color="Green" title="E56">A44</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">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="c11">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E23:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A45</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c11">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="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T12" target="_self" title="MATRLIN:th.12">Th16</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E58">A46</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c11">c</font>)</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="matrix_1.html#D2" title="MATRIX_1:def.2">MATRIX_1:def 2</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D6" target="_self" title="MATRLIN:def.6">Def8</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c11">c</font>)</span>)</span> <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> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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> )</span><br/><span class="kw">assume </span><a NAME="E1:45_5"/><span class="lab"><font color="Green" title="E59">A47</font></span>: 
<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#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c11">c</font>)</span>)</span> <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> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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></span><br/><a NAME="E2:45_5"/><span class="lab"><font color="Green" title="E60">A48</font></span>: 
<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="c11">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:45"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="txt" href="#E23:45"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="txt" href="#E24:45"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E1:45_5"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><a NAME="E3:45_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A49</font></span>: 
( <font color="Maroon" title="c11">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="c11">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c8">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="c8">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:45"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E4:45_5"/>
<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#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_5"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E5:45_5"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D6" target="_self" title="MATRLIN:def.6">Def8</a></span>;<br/><a NAME="E6:45_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A50</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A51</font></span>: <font color="Maroon" title="c12">M1</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_5"><span class="lab"><font color="Green" title="E62">A50</font></span></a>, <a class="ref" href="matrix_2.html#T16" title="MATRIX_2:th.16">MATRIX_2:16</a></span>
;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <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="c8">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#K5" title="MATRLIN:func.5">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="c10">b</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E9:45_5"/><span class="lab"><font color="Green" title="E64">A52</font></span>: 
(  <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="c10">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>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span><a NAME="E10:45_5"/><span class="lab"><font color="Green" title="E65">A53</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E9:45_5"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><span class="lab"><font color="Green" title="E66">A54</font></span>:  <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45_5"><span class="lab"><font color="Green" title="E65">A53</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_5"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E12:45_5"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">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="c10">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T12" target="_self" title="MATRLIN:th.12">Th16</a></span>;<br/><a NAME="E13:45_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A55</font></span>: 
(  <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#K5" title="MATRLIN:func.5">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="c10">b</font>)</span>)</span> &amp;  <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#K5" title="MATRLIN:func.5">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="c10">b</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_5"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><a NAME="E14:45_5"/><span class="lab"><font color="Green" title="E68">A56</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E9:45_5"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E15:45_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A57</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E16:45_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A58</font></span>: 
 <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="pre_poly.html#K9" title="PRE_POLY:func.9">/.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_5"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E17:45_5"/><span class="lab"><font color="Green" title="E71">A59</font></span>: 
 <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="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_5"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E72">A60</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c13">i</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )</span><br/><span class="kw">assume </span><a NAME="E1:45_5_3"/><span class="lab"><font color="Green" title="E73">A61</font></span>: 
<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="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c13">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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></span><br/><a NAME="E2:45_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A62</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45_5"><span class="lab"><font color="Green" title="E65">A53</font></span></a>, <a class="txt" href="#E16:45_5"><span class="lab"><font color="Green" title="E70">A58</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E3:45_5_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45_5"><span class="lab"><font color="Green" title="E68">A56</font></span></a>, <a class="txt" href="#E6:45_5"><span class="lab"><font color="Green" title="E62">A50</font></span></a>, <a class="txt" href="#E16:45_5"><span class="lab"><font color="Green" title="E70">A58</font></span></a>, <a class="txt" href="#E1:45_5_3"><span class="lab"><font color="Green" title="E73">A61</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E4:45_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A63</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></span>;<br/><a NAME="E5:45_5_3"/><span class="lab"><font color="Green" title="E76">A64</font></span>: 
<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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_5"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="txt" href="#E16:45_5"><span class="lab"><font color="Green" title="E70">A58</font></span></a>, <a class="txt" href="#E1:45_5_3"><span class="lab"><font color="Green" title="E73">A61</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:45_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A65</font></span>: 
<font color="Maroon" title="c10">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="c10">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E7:45_5_3"/><span class="lab"><font color="Green" title="E78">A66</font></span>: 
<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#K5" title="MATRLIN:func.5">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="c10">b</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_5"><span class="lab"><font color="Green" title="E66">A54</font></span></a>, <a class="txt" href="#E5:45_5_3"><span class="lab"><font color="Green" title="E76">A64</font></span></a>, <a class="ref" href="matrlin.html#T12" target="_self" title="MATRLIN:th.12">Th16</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A67</font></span>: <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">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="c10">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#K5" title="MATRLIN:func.5">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="c10">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<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="c10">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_5_3"><span class="lab"><font color="Green" title="E78">A66</font></span></a>, <a class="ref" href="funcop_1.html#T22" title="FUNCOP_1:th.22">FUNCOP_1:22</a></span>
<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="c10">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_5_3"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="txt" href="#E2:45_5_3"><span class="lab"><font color="Green" title="E74">A62</font></span></a>, <a class="ref" href="vectsp_1.html#D12" title="VECTSP_1:def.12">VECTSP_1:def 12</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45_5"><span class="lab"><font color="Green" title="E68">A56</font></span></a>, <a class="txt" href="#E7:45_5"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E16:45_5"><span class="lab"><font color="Green" title="E70">A58</font></span></a>, <a class="txt" href="#E1:45_5_3"><span class="lab"><font color="Green" title="E73">A61</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c10">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:45"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E4:45_5_3"><span class="lab"><font color="Green" title="E75">A63</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">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#K5" title="MATRLIN:func.5">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="c10">b</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45_5_3"><span class="lab"><font color="Green" title="E79">A67</font></span></a>, <a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_5"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="txt" href="#E17:45_5"><span class="lab"><font color="Green" title="E71">A59</font></span></a>, <a class="txt" href="#E16:45_5"><span class="lab"><font color="Green" title="E70">A58</font></span></a>, <a class="txt" href="#E1:45_5_3"><span class="lab"><font color="Green" title="E73">A61</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E19:45_5"/><span class="lab"><font color="Green" title="E73">A68</font></span>: 
 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="c8">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#K5" title="MATRLIN:func.5">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="c10">b</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_5"><span class="lab"><font color="Green" title="E64">A52</font></span></a></span>;<br/><a NAME="E20:45_5"/>
<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="c8">p</font>,<font color="Maroon" title="c11">c</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E1:45_5"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c11">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="c8">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="c11">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T22" title="FUNCOP_1:th.22">FUNCOP_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">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="c11">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_5"><span class="lab"><font color="Green" title="E61">A49</font></span></a>, <a class="ref" href="vectsp_1.html#D12" title="VECTSP_1:def.12">VECTSP_1:def 12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">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#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">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="c10">b</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E2:45_5"><span class="lab"><font color="Green" title="E60">A48</font></span></a></span>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c14">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:45_5"><span class="lab"><font color="Green" title="E67">A55</font></span></a>, <a class="txt" href="#E19:45_5"><span class="lab"><font color="Green" title="E73">A68</font></span></a>, <a class="ref" href="rlvect_2.html#T67" title="RLVECT_2:th.67">RLVECT_2:67</a></span>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M1</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c13">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:45_5"><span class="lab"><font color="Green" title="E69">A57</font></span></a>, <a class="txt" href="#E7:45_5"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E18:45_5"><span class="lab"><font color="Green" title="E72">A60</font></span></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_5"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="ref" href="matrlin.html#D6" target="_self" title="MATRLIN:def.6">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_5"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c11">c</font>)</span>)</span> = 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c12">M1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></span>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T32" target="_self" title="MATRLIN:th.32">Th37</a></span>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K5" title="MATRLIN:func.5">lmlt</a> (<font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">b</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="txt" href="#E21:45"><span class="lab"><font color="Green" title="E55">A19</font></span></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
