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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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><br/> for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">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="Olive" title="b1">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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; <font color="Olive" title="b3">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="Olive" title="b2">B</font> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">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="Olive" title="b2">B</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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> holds <br/> <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> <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="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">i</font></span>)</span></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="c2">A</font> &amp;  <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> 1 &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</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="finseqop.html#K4" title="FINSEQOP: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; <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="c3">B</font> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">B</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="c3">B</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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> holds <br/> <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> <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="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">i</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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="c2">A</font> &amp;  <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> 1 &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">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="finseqop.html#K4" title="FINSEQOP: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; <font color="Maroon" title="c4">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="c3">B</font> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">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="c3">B</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</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="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">i</font></span>)</span> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c4">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="c2">A</font> &amp;  <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> 1 )
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green" title="E48">A2</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">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="finseqop.html#K4" title="FINSEQOP: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>
 <b>and </b><br/><a NAME="E3:56"/><i><font color="Green" title="E49">A3</font></i>: 
( <font color="Maroon" title="c4">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="c3">B</font> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">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="c3">B</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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> <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="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">i</font></span>)</span></span><br/>

<b>thus </b><a NAME="E4:56"/>
 <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:56"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="matrix15.html#T46" target="_self" title="MATRIX15:th.46">Th46</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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></span><br/>

<b>set </b><font color="Maroon" title="c5">S</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="c2">A</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">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="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">i</font></span>)</span> or <font color="Maroon" title="c6">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="c2">A</font>,<font color="Maroon" title="c3">B</font> )</span><br/>

<b>assume </b><a NAME="E5:56"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c6">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="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><br/>

<b>consider </b><font color="Maroon" title="c7">mA</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:56"/><i><font color="Green" title="E51">A5</font></i>: 
(  <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> <font color="Maroon" title="c7">mA</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K2" title="MATRLIN:func.2">Del</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">mA</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:56"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">mB</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:56"/><i><font color="Green" title="E52">A6</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">mB</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K2" title="MATRLIN:func.2">Del</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">mB</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E3:56"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></i>;<br/>
<a NAME="E10:56"/><i><font color="Green" title="E53">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">B</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</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="c2">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix15.html#E5:56"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="matrix15.html#E7:56"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="matrix15.html#E9:56"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="matrix15.html#T33" target="_self" title="MATRIX15:th.33">Th33</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:56"/><b>then </b><i><font color="Green" title="E54">A8</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font> <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="c3">B</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">l1</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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="matrix15.html#E1:56"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<a NAME="E13:56"/><i><font color="Green" title="E55">A9</font></i>: 
(  <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <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"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></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> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E14:56"/><b>then </b><i><font color="Green" title="E56">A10</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <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="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="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">l1</font> &amp; <font color="Maroon" title="c9">l1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<b>by </b><i><a class="txt" href="matrix15.html#E1:56"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="matrix15.html#E10:56"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">j</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="c2">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <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="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="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> implies (  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c10">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="finseqop.html#K4" title="FINSEQOP: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="c3">B</font>,<font color="Maroon" title="c10">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="c3">B</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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/><b>assume </b><a NAME="E1:56_1"/><i><font color="Green" title="E57">A11</font></i>: 
<font color="Maroon" title="c10">j</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="c2">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <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="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="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c10">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="finseqop.html#K4" title="FINSEQOP: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="c3">B</font>,<font color="Maroon" title="c10">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="c3">B</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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="E2:56_1"/>
<font color="Maroon" title="c10">j</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="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <b>by </b><i><a class="txt" href="matrix15.html#E10:56"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="matrix15.html#E1:56_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="xboole_1.html#T48" title="XBOOLE_1:th.48">XBOOLE_1:48</a></i>;<br/><a NAME="E3:56_1"/><b>then </b>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:56_1"/>
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c10">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="finseqop.html#K4" title="FINSEQOP: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="c3">B</font>,<font color="Maroon" title="c10">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="c3">B</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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> )
 <b>by </b><i><a class="txt" href="matrix15.html#E2:56"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="matrix15.html#E3:56"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then </b> <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> = 
 <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="c2">A</font>,<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"><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="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></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="c3">B</font>,<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"><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="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></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>
<b>by </b><i><a class="txt" href="matrix15.html#E10:56"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="matrix15.html#E11:56"><i><font color="Green" title="E54">A8</font></i></a>, <a class="txt" href="matrix15.html#E13:56"><i><font color="Green" title="E55">A9</font></i></a>, <a class="txt" href="matrix15.html#E14:56"><i><font color="Green" title="E56">A10</font></i></a>, <a class="ref" href="matrix15.html#T45" target="_self" title="MATRIX15:th.45">Th45</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>
<br/>.= 
 <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c3">B</font>,<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"><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="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></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>
<b>by </b><i><a class="ref" href="matrix13.html#T51" title="MATRIX13:th.51">MATRIX13:51</a></i>
<br/>.= 
 <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E10:56"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="matrix13.html#T51" title="MATRIX13:th.51">MATRIX13:51</a></i>
;<br/>
<b>hence </b><a NAME="E17:56"/>
<font color="Maroon" title="c6">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="c2">A</font>,<font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E5:56"><i><font color="Green" title="E50">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
