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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">B</font> &amp; (  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) holds <br/>(  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Olive" title="b2">B</font></span>)</span> iff  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">B</font> &amp; (  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) implies (  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> iff  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:68"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E2:68"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> iff  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">wB</font> =  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">L</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">B9</font> = <font color="Maroon" title="c3">B</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">wA</font> =  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">A9</font> = <font color="Maroon" title="c2">A</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>) <span class="kw">-&gt; </span>   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> ($1,$2,<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> ($1,$2)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default">$4 <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> ($1,$3)</span>)</span></span>)</span></span>)</span>);<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">A1</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">B1</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/>(  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Olive" title="b2">B1</font></span>)</span> &amp;  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c6">B9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">B1</font>) );<br/>
<a NAME="E5:68"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
 for <font color="Olive" title="b1">A1</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">B1</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">B1</font>] holds <br/> for <font color="Olive" title="b3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">A1</font> &amp; ( <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j</font> implies <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> ) holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Olive" title="b1">A1</font>,<font color="Olive" title="b4">i</font>,<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="Olive" title="b1">A1</font>,<font color="Olive" title="b4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b1">A1</font>,<font color="Olive" title="b5">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">B1</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>,<font color="Olive" title="b3">a</font>)]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">A1</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B1</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b1">B1</font>] holds <br/> for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</font> &amp; ( <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j</font> implies <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> ) holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b3">i</font>,<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="c9">A1</font>,<font color="Olive" title="b3">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b4">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">B1</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>,<font color="Olive" title="b2">a</font>)]</span><br/><span class="kw">let </span><font color="Maroon" title="c10">B1</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">B1</font>] implies  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</font> &amp; ( <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">j</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> ) holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b2">i</font>,<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="c9">A1</font>,<font color="Olive" title="b2">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>,<font color="Olive" title="b1">a</font>)] )</span><br/>



<span class="kw">assume </span><a NAME="E1:68_1"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">B1</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</font> &amp; ( <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">j</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> ) holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b2">i</font>,<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="c9">A1</font>,<font color="Olive" title="b2">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>,<font color="Olive" title="b1">a</font>)]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</font> &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> implies <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> ) holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b1">i</font>,<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="c9">A1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c11">a</font>)]</span><br/><span class="kw">let </span><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</font> &amp; ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> implies <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> ) implies <font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<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="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>)] )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:68_1"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</font>
 <span class="kw">and </span><br/><a NAME="E3:68_1"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> implies <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<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="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>)]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c14">LAj</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c15">LAi</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c16">RA</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<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="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>);<br/>
<span class="lab"><font color="Green" title="E61">A7</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</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="c9">A1</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D2" title="MATRIX_1:def.2">MATRIX_1:def 2</a></span>
;<br/>
<a NAME="E5:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A8</font></span>: 
 <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">B1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>))
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68_1"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E3:68_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="ref" href="matrix15.html#T40" target="_self" title="MATRIX15:th.40">Th40</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">RB</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c18">LBj</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c13">j</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c19">LBi</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>);<br/>
<a NAME="E6:68_1"/><span class="lab"><font color="Green" title="E63">A9</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D2" title="MATRIX_1:def.2">MATRIX_1:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2"><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="E1:68_1_2"/>
(  not <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span> or <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:68_1_2_1"/>
 not <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<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="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>)]</span><br/><div class="add"><a NAME="E2:68_1_2_1"/><span class="kw">then </span>
(  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<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="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="ref" href="matrix13.html#T40" title="MATRIX13:th.40">MATRIX13:40</a></span>;<br/><span class="kw">hence </span><a NAME="E3:68_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<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="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>)]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E58">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:68_1_2_2"/><span class="lab"><font color="Green" title="E64">A10</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<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="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>)]</span><br/><div class="add"><a NAME="E2:68_1_2_2"/><span class="lab"><font color="Green" title="E65">A11</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="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D2" title="MATRIX_1:def.2">MATRIX_1:def 2</a></span>;<br/><a NAME="E3:68_1_2_2"/><span class="lab"><font color="Green" title="E66">A12</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="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">A1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c10">B1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E4:68_1_2_2"/><span class="lab"><font color="Green" title="E67">A13</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">A1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c10">B1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E5:68_1_2_2"/>
(  <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="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">A1</font> &amp;  <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="c10">B1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c10">B1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>) = 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c12">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T18" target="_self" title="MATRIX15:th.18">Th18</a></span>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c12">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1_2_2"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="txt" href="#E3:68_1_2_2"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="ref" href="matrix15.html#T3" target="_self" title="MATRIX15:th.3">Th3</a></span>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c12">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T4" target="_self" title="MATRIX15:th.4">Th4</a></span>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c12">i</font>,<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> (<span class="p3">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1_2_2"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="ref" href="matrix15.html#T15" target="_self" title="MATRIX15:th.15">Th15</a></span>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c12">i</font>,<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> (<span class="p3">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68_1"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="matrix15.html#T15" target="_self" title="MATRIX15:th.15">Th15</a></span>
;<br/><a NAME="E7:68_1_2_2"/><span class="kw">then </span>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68_1"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E3:68_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E2:68_1_2_2"><span class="lab"><font color="Green" title="E65">A11</font></span></a>, <a class="ref" href="matrix13.html#T92" title="MATRIX13:th.92">MATRIX13:92</a></span>;<br/><span class="kw">hence </span><a NAME="E8:68_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c12">i</font>,<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="c9">A1</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span>)</span>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>,<font color="Maroon" title="c11">a</font>)]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E5:68_1"><span class="lab"><font color="Green" title="E62">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E6:68"/><span class="lab"><font color="Green" title="E58">A14</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c6">B9</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">A1</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c10">B1</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c11">N</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a><span class="kw"> such that </span><br/><a NAME="E8:68"/><span class="lab"><font color="Green" title="E59">A15</font></span>: 
<font color="Maroon" title="c11">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:68"/><span class="lab"><font color="Green" title="E60">A16</font></span>: 
(  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c8">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c9">A1</font> &amp;  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c8">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font> )
 <span class="kw">and </span><br/><a NAME="E10:68"/><span class="lab"><font color="Green" title="E61">A17</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">B1</font>] &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c9">A1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">N</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span>) )
 <span class="kw">and </span><br/><a NAME="E11:68"/><span class="lab"><font color="Green" title="E62">A18</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b1">i</font>) <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="c2">A</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:68"/>
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">A1</font></span>)</span> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c11">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c9">A1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="matrix15.html#S2" target="_self" title="MATRIX15:sch.2">MATRIX15:sch 2</a>(<a class="txt" href="#E6:68"><span class="lab"><font color="Green" title="E58">A14</font></span></a>, <a class="txt" href="#E5:68"><span class="lab"><font color="Green" title="E57">A3</font></span></a>);<br/></span>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2"><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="E1:68_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</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  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</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="68_2_1"><span class="kw">suppose </span></a><a NAME="E1:68_2_1"/><span class="lab"><font color="Green" title="E63">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> iff  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/><div class="add"><a NAME="E2:68_2_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T74" title="MATRIX13:th.74">MATRIX13:74</a>, <a class="ref" href="matrix_1.html#D2" title="MATRIX_1:def.2">MATRIX_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:68_2_1"/>
(  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> iff  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_1"><span class="lab"><font color="Green" title="E63">A19</font></span></a>, <a class="ref" href="matrix15.html#T51" target="_self" title="MATRIX15:th.51">Th51</a>, <a class="ref" href="matrix13.html#T74" title="MATRIX13:th.74">MATRIX13:74</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2"><span class="kw">suppose </span></a><a NAME="E1:68_2_2"/><span class="lab"><font color="Green" title="E63">A20</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> iff  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_1"><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="E1:68_2_2_1"/>
( <font color="Maroon" title="c11">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c11">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:68_2_2_1_1"/><span class="lab"><font color="Green" title="E64">A21</font></span>: 
<font color="Maroon" title="c11">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> iff  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c12">SN</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">SS</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>;<br/><a NAME="E2:68_2_2_1_1"/><span class="lab"><font color="Green" title="E65">A22</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">P2</font> = <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>, <font color="Maroon" title="c15">Q2</font> = <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c11">N</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="matrix13.html#T53" title="MATRIX13:th.53">MATRIX13:53</a></span>;<br/><a NAME="E4:68_2_2_1_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span></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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span></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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a>, <a class="ref" href="finseq_3.html#T40" title="FINSEQ_3:th.40">FINSEQ_3:40</a></span>;<br/><a NAME="E5:68_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A23</font></span>: 
<font color="Maroon" title="c14">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68_2_2_1_1"><span class="lab"><font color="Green" title="E65">A22</font></span></a>, <a class="ref" href="relat_1.html#T134" title="RELAT_1:th.134">RELAT_1:134</a></span>;<br/><a NAME="E6:68_2_2_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></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_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><a NAME="E7:68_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A24</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c15">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="ref" href="funct_1.html#T77" title="FUNCT_1:th.77">FUNCT_1:77</a></span>;<br/><a NAME="E8:68_2_2_1_1"/>
( <font color="Maroon" title="c15">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T92" title="FINSEQ_3:th.92">FINSEQ_3:92</a>, <a class="ref" href="relat_1.html#T132" title="RELAT_1:th.132">RELAT_1:132</a></span>;<br/><a NAME="E9:68_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">N</font>,<font color="Maroon" title="c15">Q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68_2_2_1_1"><span class="lab"><font color="Green" title="E67">A24</font></span></a>, <a class="ref" href="card_1.html#T33" title="CARD_1:th.33">CARD_1:33</a></span>;<br/><a NAME="E10:68_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A25</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">Q2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T5" title="CARD_1:th.5">CARD_1:5</a>, <a class="ref" href="finseq_3.html#T40" title="FINSEQ_3:th.40">FINSEQ_3:40</a></span>;<br/><a NAME="E11:68_2_2_1_1"/><span class="lab"><font color="Green" title="E69">A26</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">A1</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> <font color="Maroon" title="c9">A1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E12:68_2_2_1_1"/><span class="lab"><font color="Green" title="E70">A27</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><div><span class="lab"><font color="Green" title="E71">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span> implies  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c16">i</font>) <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="c9">A1</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:68_2_2_1_1_1"/><span class="lab"><font color="Green" title="E72">A29</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c16">i</font>) <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="c9">A1</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span><br/><a NAME="E2:68_2_2_1_1_1"/>
 not <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2_1_1_1"><span class="lab"><font color="Green" title="E72">A29</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:68_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A30</font></span>: 
( <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2_1_1_1"><span class="lab"><font color="Green" title="E72">A29</font></span></a></span>;<br/><a NAME="E4:68_2_2_1_1_1"/>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2_1_1_1"><span class="lab"><font color="Green" title="E72">A29</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:68_2_2_1_1_1"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c16">i</font>) <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="c9">A1</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="txt" href="#E11:68_2_2_1_1"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E12:68_2_2_1_1"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="#E3:68_2_2_1_1_1"><span class="lab"><font color="Green" title="E73">A30</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:68_2_2_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="ref" href="matrix13.html#T74" title="MATRIX13:th.74">MATRIX13:74</a></span>;<br/><a NAME="E15:68_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A31</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <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="E16:68_2_2_1_1"/><span class="lab"><font color="Green" title="E73">A32</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E17:68_2_2_1_1"/><span class="lab"><font color="Green" title="E74">A33</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">B1</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> <font color="Maroon" title="c10">B1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c10">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><span class="kw">thus </span><a NAME="E18:68_2_2_1_1"/>
(  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> implies  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:68_2_2_1_1_2"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<a NAME="E2:68_2_2_1_1_2"/><span class="kw">then </span>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E61">A17</font></span></a></span>;<br/>
<a NAME="E3:68_2_2_1_1_2"/><span class="kw">then </span>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span> holds <br/>(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Olive" title="b1">i</font>) <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="c9">A1</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Olive" title="b1">i</font>) <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="c10">B1</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68_2_2_1_1"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E16:68_2_2_1_1"><span class="lab"><font color="Green" title="E73">A32</font></span></a>, <a class="txt" href="#E13:68_2_2_1_1"><span class="lab"><font color="Green" title="E71">A28</font></span></a>, <a class="ref" href="matrix15.html#T24" target="_self" title="MATRIX15:th.24">Th24</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E4:68_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A34</font></span>: 
 <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">B1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c9">A1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c10">B1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2_1_1"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="txt" href="#E15:68_2_2_1_1"><span class="lab"><font color="Green" title="E72">A31</font></span></a>, <a class="txt" href="#E11:68_2_2_1_1"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E16:68_2_2_1_1"><span class="lab"><font color="Green" title="E73">A32</font></span></a>, <a class="txt" href="#E12:68_2_2_1_1"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="#E17:68_2_2_1_1"><span class="lab"><font color="Green" title="E74">A33</font></span></a>, <a class="ref" href="matrix15.html#T45" target="_self" title="MATRIX15:th.45">Th45</a></span>;<br/>
<a NAME="E5:68_2_2_1_1_2"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c9">A1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c14">P2</font>,<font color="Maroon" title="c15">Q2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="ref" href="matrix13.html#T56" title="MATRIX13:th.56">MATRIX13:56</a></span>;<br/>
<a NAME="E6:68_2_2_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">X</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">K</font> st <br/>(  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Olive" title="b1">X</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c15">Q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span>) &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c15">Q2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c10">B1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span>) &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c9">A1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c10">B1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span>)</span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2_1_1"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="txt" href="#E2:68_2_2_1_1"><span class="lab"><font color="Green" title="E65">A22</font></span></a>, <a class="txt" href="#E5:68_2_2_1_1"><span class="lab"><font color="Green" title="E66">A23</font></span></a>, <a class="txt" href="#E10:68_2_2_1_1"><span class="lab"><font color="Green" title="E68">A25</font></span></a>, <a class="ref" href="matrix15.html#T50" target="_self" title="MATRIX15:th.50">Th50</a>, <a class="ref" href="relat_1.html#T132" title="RELAT_1:th.132">RELAT_1:132</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:68_2_2_1_1_2"/>
 not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="txt" href="#E4:68_2_2_1_1_2"><span class="lab"><font color="Green" title="E75">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E19:68_2_2_1_1"/><span class="lab"><font color="Green" title="E75">A35</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><span class="kw">thus </span><a NAME="E20:68_2_2_1_1"/>
(  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:68_2_2_1_1_3"/>
 not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span></span><br/>

<a NAME="E2:68_2_2_1_1_3"/><span class="kw">then </span>
 not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">B1</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E61">A17</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:68_2_2_1_1_3"/><span class="lab"><font color="Green" title="E76">A36</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">B1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">AB</font> = <font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font>;<br/>
<a NAME="E5:68_2_2_1_1_3"/><span class="lab"><font color="Green" title="E77">A37</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D2" title="MATRIX_1:def.2">MATRIX_1:def 2</a></span>;<br/>
<a NAME="E6:68_2_2_1_1_3"/><span class="lab"><font color="Green" title="E78">A38</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="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c16">x</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="txt" href="#E4:68_2_2_1_1_3"><span class="lab"><font color="Green" title="E76">A36</font></span></a>, <a class="ref" href="matrix15.html#T53" target="_self" title="MATRIX15:th.53">Th53</a></span>;<br/>
<a NAME="E8:68_2_2_1_1_3"/><span class="lab"><font color="Green" title="E79">A39</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">A1</font></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="ref" href="matrix15.html#T19" target="_self" title="MATRIX15:th.19">Th19</a></span>;<br/>
<a NAME="E9:68_2_2_1_1_3"/><span class="lab"><font color="Green" title="E80">A40</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</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="c9">A1</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="c10">B1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span> implies  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c19">i</font>) <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> <span class="p2">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:68_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E81">A41</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c19">i</font>) <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> <span class="p2">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span><br/><a NAME="E2:68_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E82">A42</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c19">i</font>) <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="c9">A1</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:68_2_2_1_1"><span class="lab"><font color="Green" title="E71">A28</font></span></a>, <a class="txt" href="#E1:68_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E81">A41</font></span></a></span>;<br/><a NAME="E3:68_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E83">A43</font></span>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">B1</font>) &amp; <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</font> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c19">i</font>) <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="c9">A1</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> implies  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c19">i</font>) <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="c10">B1</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T41" target="_self" title="MATRIX15:th.41">Th41</a></span>;<br/><span class="kw">thus </span> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<font color="Maroon" title="c19">i</font>) = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c19">i</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="c10">B1</font>,<font color="Maroon" title="c19">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:68_2_2_1_1"><span class="lab"><font color="Green" title="E75">A35</font></span></a>, <a class="txt" href="#E1:68_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E81">A41</font></span></a>, <a class="ref" href="matrix15.html#T15" target="_self" title="MATRIX15:th.15">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68_2_2_1_1"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E19:68_2_2_1_1"><span class="lab"><font color="Green" title="E75">A35</font></span></a>, <a class="txt" href="#E4:68_2_2_1_1_3"><span class="lab"><font color="Green" title="E76">A36</font></span></a>, <a class="txt" href="#E9:68_2_2_1_1_3"><span class="lab"><font color="Green" title="E80">A40</font></span></a>, <a class="txt" href="#E1:68_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E81">A41</font></span></a>, <a class="txt" href="#E2:68_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E82">A42</font></span></a>, <a class="txt" href="#E3:68_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E83">A43</font></span></a>, <a class="ref" href="finseq_2.html#T123" title="FINSEQ_2:th.123">FINSEQ_2:123</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:68_2_2_1_1_3"/><span class="kw">then </span>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:68_2_2_1_1"><span class="lab"><font color="Green" title="E72">A31</font></span></a>, <a class="txt" href="#E6:68_2_2_1_1_3"><span class="lab"><font color="Green" title="E78">A38</font></span></a>, <a class="ref" href="matrix15.html#T11" target="_self" title="MATRIX15:th.11">Th11</a></span>;<br/>
<a NAME="E12:68_2_2_1_1_3"/><span class="kw">then </span>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_2_2_1_1_3"><span class="lab"><font color="Green" title="E77">A37</font></span></a>, <a class="ref" href="matrix13.html#T74" title="MATRIX13:th.74">MATRIX13:74</a></span>;<br/>
<a NAME="E13:68_2_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A44</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/>
<a NAME="E14:68_2_2_1_1_3"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">A1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68_2_2_1_1_3"><span class="lab"><font color="Green" title="E80">A40</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E15:68_2_2_1_1_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span></span>)</span>
 
<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="E16:68_2_2_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">A1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_2_2_1_1_3"><span class="lab"><font color="Green" title="E78">A38</font></span></a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<a NAME="E17:68_2_2_1_1_3"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68_2_2_1_1_3"><span class="lab"><font color="Green" title="E79">A39</font></span></a>, <a class="ref" href="matrix13.html#T79" title="MATRIX13:th.79">MATRIX13:79</a></span>;<br/>
<a NAME="E18:68_2_2_1_1_3"/><span class="kw">then </span>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:68_2_2_1_1_3"><span class="lab"><font color="Green" title="E81">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:68_2_2_1_1_3"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E61">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:68_2_2_1_2"/><span class="lab"><font color="Green" title="E64">A45</font></span>: 
<font color="Maroon" title="c11">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> iff  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c12">ZERO</font> =  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>);<br/><div><span class="lab"><font color="Green" title="E65">A46</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> implies <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:68_2_2_1_2_1"/><span class="lab"><font color="Green" title="E66">A47</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span><br/><a NAME="E2:68_2_2_1_2_1"/><span class="lab"><font color="Green" title="E67">A48</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">A1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</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>, <a class="ref" href="matrix_1.html#D2" title="MATRIX_1:def.2">MATRIX_1:def 2</a></span>;<br/><a NAME="E3:68_2_2_1_2_1"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E4:68_2_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A49</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2_1_2_1"><span class="lab"><font color="Green" title="E66">A47</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></span>
<br/>.= 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c13">i</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="txt" href="#E1:68_2_2_1_2"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="txt" href="#E4:68_2_2_1_2_1"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E2:68_2_2_1_2_1"><span class="lab"><font color="Green" title="E67">A48</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_2_2_1_2_1"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="ref" href="matrix_2.html#T8" title="MATRIX_2:th.8">MATRIX_2:8</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:68_2_2_1_2"/><span class="lab"><font color="Green" title="E66">A50</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E4:68_2_2_1_2"/><span class="lab"><font color="Green" title="E67">A51</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E5:68_2_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E6:68_2_2_1_2"/><span class="kw">then </span>
 <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_2_2_1_2"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="txt" href="#E2:68_2_2_1_2"><span class="lab"><font color="Green" title="E65">A46</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/><a NAME="E7:68_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A52</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E3:68_2_2_1_2"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="txt" href="#E4:68_2_2_1_2"><span class="lab"><font color="Green" title="E67">A51</font></span></a>, <a class="ref" href="matrix13.html#T95" title="MATRIX13:th.95">MATRIX13:95</a></span>;<br/><a NAME="E8:68_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A53</font></span>: 
 <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T95" title="MATRIX13:th.95">MATRIX13:95</a></span>;<br/><a NAME="E9:68_2_2_1_2"/><span class="lab"><font color="Green" title="E70">A54</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><span class="kw">thus </span><a NAME="E10:68_2_2_1_2"/>
(  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> implies  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c13">x</font> =  the   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font>;<br/>
<span class="kw">assume </span><a NAME="E1:68_2_2_1_2_2"/><span class="lab"><font color="Green" title="E71">A55</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<a NAME="E2:68_2_2_1_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E3:68_2_2_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A56</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68_2_2_1_2"><span class="lab"><font color="Green" title="E70">A54</font></span></a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<a NAME="E4:68_2_2_1_2_2"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T19" target="_self" title="MATRIX15:th.19">Th19</a></span>;<br/>
<a NAME="E5:68_2_2_1_2_2"/><span class="kw">then </span>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68_2_2_1_2"><span class="lab"><font color="Green" title="E68">A52</font></span></a>, <a class="txt" href="#E1:68_2_2_1_2_2"><span class="lab"><font color="Green" title="E71">A55</font></span></a>, <a class="txt" href="#E3:68_2_2_1_2_2"><span class="lab"><font color="Green" title="E72">A56</font></span></a>, <a class="ref" href="matrix13.html#T79" title="MATRIX13:th.79">MATRIX13:79</a></span>;<br/>
<a NAME="E6:68_2_2_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A57</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="matrix13.html#T95" title="MATRIX13:th.95">MATRIX13:95</a></span>;<br/>
<a NAME="E7:68_2_2_1_2_2"/><span class="kw">then </span>
( (  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</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  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c6">B9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font> : verum </span> } </span>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="txt" href="#E8:68_2_2_1_2"><span class="lab"><font color="Green" title="E69">A53</font></span></a>, <a class="ref" href="matrix15.html#T54" target="_self" title="MATRIX15:th.54">Th54</a></span>;<br/>
<a NAME="E8:68_2_2_1_2_2"/><span class="kw">then </span>
(  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c6">B9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> or  the   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c6">B9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68_2_2_1_2"><span class="lab"><font color="Green" title="E69">A53</font></span></a>, <a class="txt" href="#E6:68_2_2_1_2_2"><span class="lab"><font color="Green" title="E73">A57</font></span></a>, <a class="ref" href="matrix15.html#T56" target="_self" title="MATRIX15:th.56">Th56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:68_2_2_1_2_2"/>
 not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E11:68_2_2_1_2"/><span class="lab"><font color="Green" title="E71">A58</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">B9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E12:68_2_2_1_2"/><span class="lab"><font color="Green" title="E72">A59</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">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="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E13:68_2_2_1_2"/><span class="lab"><font color="Green" title="E73">A60</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_2"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><span class="kw">thus </span><a NAME="E14:68_2_2_1_2"/>
(  not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:68_2_2_1_2_3"/><span class="lab"><font color="Green" title="E74">A61</font></span>: 
 not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span></span><br/>

<span class="kw">assume </span><a NAME="E2:68_2_2_1_2_3"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <font color="Maroon" title="c3">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:68_2_2_1_2_3"/><span class="lab"><font color="Green" title="E75">A62</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:68_2_2_1_2_3"/><span class="lab"><font color="Green" title="E76">A63</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68_2_2_1_2"><span class="lab"><font color="Green" title="E68">A52</font></span></a>, <a class="ref" href="matrix13.html#T94" title="MATRIX13:th.94">MATRIX13:94</a></span>;<br/>
<a NAME="E6:68_2_2_1_2_3"/><span class="lab"><font color="Green" title="E77">A64</font></span>: 
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68_2_2_1_2"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E4:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E7:68_2_2_1_2_3"/><span class="lab"><font color="Green" title="E78">A65</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68_2_2_1_2"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="ref" href="finseq_2.html#T124" title="FINSEQ_2:th.124">FINSEQ_2:124</a></span>;<br/>
<a NAME="E8:68_2_2_1_2_3"/><span class="lab"><font color="Green" title="E79">A66</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E9:68_2_2_1_2_3"/><span class="lab"><font color="Green" title="E80">A67</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">B9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T124" title="FINSEQ_2:th.124">FINSEQ_2:124</a></span>;<br/>
<a NAME="E10:68_2_2_1_2_3"/><span class="lab"><font color="Green" title="E81">A68</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T124" title="FINSEQ_2:th.124">FINSEQ_2:124</a></span>;<br/>
<a NAME="E11:68_2_2_1_2_3"/><span class="lab"><font color="Green" title="E82">A69</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68_2_2_1_2"><span class="lab"><font color="Green" title="E70">A54</font></span></a>, <a class="txt" href="#E4:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E12:68_2_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A70</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span>,<font color="Maroon" title="c13">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c13">i</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">B9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T15" target="_self" title="MATRIX15:th.15">Th15</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_1_2_3_1"><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="E1:68_2_2_1_2_3_1"/>
( <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN: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_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> or  ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN: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_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">B9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> &amp; <font color="Maroon" title="c14">j</font> <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="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E77">A64</font></span></a>, <a class="txt" href="#E8:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E79">A66</font></span></a>, <a class="txt" href="#E7:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E78">A65</font></span></a>, <a class="txt" href="#E12:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E83">A70</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:68_2_2_1_2_3_1_1"/><span class="lab"><font color="Green" title="E84">A71</font></span>: 
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN: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_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:68_2_2_1_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A72</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:68_2_2_1_2"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E11:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E82">A69</font></span></a>, <a class="txt" href="#E10:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E81">A68</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>) = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68_2_2_1_2"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E6:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E77">A64</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c8">A9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E83">A70</font></span></a>, <a class="txt" href="#E1:68_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E84">A71</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c8">A9</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E81">A68</font></span></a>, <a class="txt" href="#E1:68_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E84">A71</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68_2_2_1_2"><span class="lab"><font color="Green" title="E69">A53</font></span></a>, <a class="txt" href="#E2:68_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E85">A72</font></span></a>, <a class="ref" href="matrix_3.html#T1" title="MATRIX_3:th.1">MATRIX_3:1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:68_2_2_1_2_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E76">A63</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:68_2_2_1_2_3_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN: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_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">B9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> &amp; <font color="Maroon" title="c14">j</font> <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="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:68_2_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E84">A73</font></span>: 
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN: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_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">B9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:68_2_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E85">A74</font></span>: 
<font color="Maroon" title="c14">j</font> <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="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">n</font>
 ;<br/><a NAME="E5:68_2_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E86">A75</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c15">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68_2_2_1_2"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="txt" href="#E11:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E82">A69</font></span></a>, <a class="txt" href="#E9:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="txt" href="#E3:68_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E84">A73</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E6:68_2_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E87">A76</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68_2_2_1_2"><span class="lab"><font color="Green" title="E69">A53</font></span></a>, <a class="txt" href="#E1:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="matrix15.html#T52" target="_self" title="MATRIX15:th.52">Th52</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>) = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">A9</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B9</font></span>)</span>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68_2_2_1_2"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E6:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E77">A64</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">B9</font>,<font color="Maroon" title="c13">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E79">A66</font></span></a>, <a class="txt" href="#E12:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E83">A70</font></span></a>, <a class="txt" href="#E3:68_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E84">A73</font></span></a>, <a class="txt" href="#E4:68_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E85">A74</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c6">B9</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c15">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="txt" href="#E3:68_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E84">A73</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E86">A75</font></span></a>, <a class="txt" href="#E6:68_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E87">A76</font></span></a>, <a class="ref" href="matrix_3.html#T1" title="MATRIX_3:th.1">MATRIX_3:1</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:68_2_2_1_2_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_2_2_1_2_3"><span class="lab"><font color="Green" title="E76">A63</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
