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<span class="kw">reconsider </span><font color="Maroon" title="c7">N</font> = <font color="Maroon" title="c2">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">AB</font> = <font color="Maroon" title="c5">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c6">B</font>;<br/>
<span class="lab"><font color="Green" title="E17">A1</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">B</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></span>
;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> = 
 <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">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E18">A2</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">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c6">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="pre_poly.html#D4" title="PRE_POLY:def.4">PRE_POLY:def 4</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>

;<br/>
<a NAME="E5:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A3</font></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">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_1_1_4"/>
( <font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c7">N</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_1_4_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">A</font> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <font color="Maroon" title="c6">B</font> is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">B</font></span>)</span>,<font color="Maroon" title="c1">D</font></span><br/><div class="add"><a NAME="E2:17_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_1_1_4_1"/>
<font color="Maroon" title="c5">A</font> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <font color="Maroon" title="c6">B</font> is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">B</font></span>)</span>,<font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_4_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="matrix_1.html#T13" title="MATRIX_1:th.13">MATRIX_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_1_4_2"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c7">N</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">A</font> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <font color="Maroon" title="c6">B</font> is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">B</font></span>)</span>,<font color="Maroon" title="c1">D</font></span><br/><div class="add"><a NAME="E2:17_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c7">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/><a NAME="E3:17_1_1_4_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">N</font> &amp; <font color="Maroon" title="c6">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">N</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/><a NAME="E4:17_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">N</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E2:17_1_1_4_2"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="pre_poly.html#D4" title="PRE_POLY:def.4">PRE_POLY:def 4</a></span>;<br/><a NAME="E5:17_1_1_4_2"/><span class="lab"><font color="Green" title="E23">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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">N</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/><a NAME="E6:17_1_1_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E2:17_1_1_4_2"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E7:17_1_1_4_2"/><span class="kw">then </span>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E1:17_1_1_4_2"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:17_1_1_4_2"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E5:17_1_1_4_2"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:17_1_1_4_2"/>
<font color="Maroon" title="c5">A</font> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <font color="Maroon" title="c6">B</font> is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">B</font></span>)</span>,<font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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