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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>, <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> for <font color="Olive" title="b4">M1</font> being   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>,<font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b5">M2</font> being   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">k</font>,<font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Olive" title="b4">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b5">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b5">M2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  for <font color="Olive" title="b1">M1</font> being   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">M2</font> being   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Olive" title="b1">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b2">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b2">M2</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10"/><span class="lab"><font color="Green" title="E9">A1</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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:10"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M1</font> being   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">M2</font> being   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Olive" title="b1">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b2">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b2">M2</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">M1</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M2</font> being   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b1">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b1">M2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">M2</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span><br/>



<span class="lab"><font color="Green" title="E11">A3</font></span>:  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font> = 
<font color="Maroon" title="c3">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>
;<br/>
<span class="lab"><font color="Green" title="E12">A4</font></span>:  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> = 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c3">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>
;<br/>
<a NAME="E5:10"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0: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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) )</span><br/>

<a NAME="E1:10_3"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</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"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</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"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</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> <font color="Maroon" title="c5">M1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_3"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D1" onmouseover="rs('polynom1/D1')" onmouseout="rh()">POLYNOM1:def 1</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> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E16">A8</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T6" onmouseover="rs('finseq_1/T6')" onmouseout="rh()">FINSEQ_1:6</a></span>
<br/>.= 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
;<br/>
<span class="lab"><font color="Green" title="E17">A9</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E7:10_3"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>)</span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E19">A11</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) = 
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
 <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> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#D9" onmouseover="rs('fvsum_1/D9')" onmouseout="rh()">FVSUM_1:def 9</a></span>
;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> = 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
;<br/>
<a NAME="E10:10_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">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="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:10_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A12</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_3"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E12:10_3"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_8"><span class="kw">proof </span></a><div class="add">

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>
;<br/>
<a NAME="E2:10_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A13</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:10_3_8"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
(  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <font color="Maroon" title="c7">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E7:10_3"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E3:10_3"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:10_3_8"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/>

<span class="lab"><font color="Green" title="E24">A16</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> = 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
;<br/>
<a NAME="E6:10_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="tarski.html#R2" title="TARSKI: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_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3_8"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:10_3_8"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/>
<a NAME="E8:10_3_8"/><span class="kw">then </span>
<font color="Maroon" title="c9">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_3_8"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:10_3_8"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">a</font> = <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E11:10_3_8"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M1</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_3_8"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/>
<a NAME="E12:10_3_8"/>
<font color="Maroon" title="c9">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E6:10_3_8"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>;<br/>
<a NAME="E13:10_3_8"/><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="c7">i</font>,<font color="Maroon" title="c9">k</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_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c5">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_3_8"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E2:10_3_8"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E14:10_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_3_8"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>;<br/>
<a NAME="E15:10_3_8"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">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="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_3_8"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E16:10_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="tarski.html#R2" title="TARSKI: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_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3_8"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>;<br/>
<a NAME="E17:10_3_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_3"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E4:10_3_8"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="fvsum_1.html#T50" onmouseover="rs('fvsum_1/T50')" onmouseout="rh()">FVSUM_1:50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:10_3_8"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_3_8"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E14:10_3_8"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E13:10_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A21</font></span>: 
 <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_3"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="finseq_1.html#T13" onmouseover="rs('finseq_1/T13')" onmouseout="rh()">FINSEQ_1:13</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D1" onmouseover="rs('polynom1/D1')" onmouseout="rh()">POLYNOM1:def 1</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> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E22">A22</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T6" onmouseover="rs('finseq_1/T6')" onmouseout="rh()">FINSEQ_1:6</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>
;<br/>
<span class="lab"><font color="Green" title="E23">A23</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></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"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">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> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:10_3"><span class="lab"><font color="Green" title="E22">A22</font></span></a>, <a class="ref" href="matrix_3.html#T6" onmouseover="rs('matrix_3/T6')" onmouseout="rh()">MATRIX_3:6</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<span class="lab"><font color="Green" title="E24">A24</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D1" onmouseover="rs('polynom1/D1')" onmouseout="rh()">POLYNOM1:def 1</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> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">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> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_3"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="matrix_3.html#T6" onmouseover="rs('matrix_3/T6')" onmouseout="rh()">MATRIX_3:6</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E18:10_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A25</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="ref" href="polynom1.html#D1" onmouseover="rs('polynom1/D1')" onmouseout="rh()">POLYNOM1:def 1</a></span>;<br/>
<a NAME="E19:10_3"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_13"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E26">A26</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>
;<br/>
<a NAME="E2:10_3_13"/><span class="lab"><font color="Green" title="E27">A27</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:10_3_13"/><span class="lab"><font color="Green" title="E28">A28</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/>

 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">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> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_3_13"><span class="lab"><font color="Green" title="E26">A26</font></span></a>, <a class="ref" href="matrix_3.html#T6" onmouseover="rs('matrix_3/T6')" onmouseout="rh()">MATRIX_3:6</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E5:10_3_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A29</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="tarski.html#R2" title="TARSKI: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_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">M1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="txt" href="#E17:10_3"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E3:10_3_13"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="txt" href="#E2:10_3_13"><span class="lab"><font color="Green" title="E27">A27</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:10_3_13"/><span class="kw">then </span>
<font color="Maroon" title="c9">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">M2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:10_3_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A30</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M2</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/>
<a NAME="E8:10_3_13"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M1</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_3_13"><span class="lab"><font color="Green" title="E29">A29</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">c</font> = <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>, <font color="Maroon" title="c11">d</font> = <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_3_13"><span class="lab"><font color="Green" title="E30">A30</font></span></a></span>;<br/>
<a NAME="E10:10_3_13"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="txt" href="#E17:10_3"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E3:10_3_13"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="ref" href="fvsum_1.html#T60" onmouseover="rs('fvsum_1/T60')" onmouseout="rh()">FVSUM_1:60</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">a</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E12:10_3_13"/><span class="lab"><font color="Green" title="E31">A31</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:10_3"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="txt" href="#E3:10_3_13"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="ref" href="fvsum_1.html#T50" onmouseover="rs('fvsum_1/T50')" onmouseout="rh()">FVSUM_1:50</a></span>;<br/>
<a NAME="E13:10_3_13"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="txt" href="#E3:10_3_13"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="ref" href="fvsum_1.html#T50" onmouseover="rs('fvsum_1/T50')" onmouseout="rh()">FVSUM_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">b</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<font color="Maroon" title="c13">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="txt" href="#E3:10_3_13"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="ref" href="fvsum_1.html#T50" onmouseover="rs('fvsum_1/T50')" onmouseout="rh()">FVSUM_1:50</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">d</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="txt" href="#E17:10_3"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E3:10_3_13"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="ref" href="fvsum_1.html#T60" onmouseover="rs('fvsum_1/T60')" onmouseout="rh()">FVSUM_1:60</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:10_3_13"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_3_13"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="txt" href="#E12:10_3_13"><span class="lab"><font color="Green" title="E31">A31</font></span></a>, <a class="ref" href="fvsum_1.html#T60" onmouseover="rs('fvsum_1/T60')" onmouseout="rh()">FVSUM_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E20:10_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A32</font></span>: 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:10_3"><span class="lab"><font color="Green" title="E21">A21</font></span></a>, <a class="txt" href="#E18:10_3"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="ref" href="finseq_1.html#T13" onmouseover="rs('finseq_1/T13')" onmouseout="rh()">FINSEQ_1:13</a></span>;<br/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c6">M2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
;<br/>
<a NAME="E22:10_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A33</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_3"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E3:10_3"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) = 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E22:10_3"><span class="lab"><font color="Green" title="E27">A33</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#D9" onmouseover="rs('fvsum_1/D9')" onmouseout="rh()">FVSUM_1:def 9</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E24:10_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:10_3"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E20:10_3"><span class="lab"><font color="Green" title="E26">A32</font></span></a>, <a class="ref" href="fvsum_1.html#T73" onmouseover="rs('fvsum_1/T73')" onmouseout="rh()">FVSUM_1:73</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="lab"><font color="Green" title="E14">A34</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
;<br/>
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> = 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
<br/>.= 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c6">M2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
;<br/>
<a NAME="E8:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A35</font></span>: 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</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"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:10"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10"><span class="lab"><font color="Green" title="E14">A34</font></span></a>, <a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E15">A35</font></span></a>, <a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="matrix_0.html#T21" onmouseover="rs('matrix_0/T21')" onmouseout="rh()">MATRIX_0:21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
