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<span class="kw">let </span><font color="Maroon" title="c1">n9</font>, <font color="Maroon" title="c2">m9</font>, <font color="Maroon" title="c3">j</font>, <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font><br/>  for <font color="Olive" title="b3">M9</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n9</font>,<font color="Maroon" title="c2">m9</font>,<font color="Olive" title="b1">K</font>  st <font color="Maroon" title="c3">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">M9</font></span>)</span> &amp; ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="Olive" title="b1">K</font></span>)</span> ) holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b3">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Olive" title="b3">M9</font>,<font color="Maroon" title="c4">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="Olive" title="b3">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">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="Olive" title="b3">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">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> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">K</font><br/>  for <font color="Olive" title="b2">M9</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n9</font>,<font color="Maroon" title="c2">m9</font>,<font color="Maroon" title="c5">K</font>  st <font color="Maroon" title="c3">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">M9</font></span>)</span> &amp; ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c5">K</font></span>)</span> ) holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b2">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Olive" title="b2">M9</font>,<font color="Maroon" title="c4">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="Olive" title="b2">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">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="Olive" title="b2">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">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">M9</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n9</font>,<font color="Maroon" title="c2">m9</font>,<font color="Maroon" title="c5">K</font>  st <font color="Maroon" title="c3">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M9</font></span>)</span> &amp; ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> implies <font color="Maroon" title="c6">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="c5">K</font></span>)</span> ) holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b1">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Olive" title="b1">M9</font>,<font color="Maroon" title="c4">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="Olive" title="b1">M9</font>,<font color="Maroon" title="c4">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="c6">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="Olive" title="b1">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">M9</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n9</font>,<font color="Maroon" title="c2">m9</font>,<font color="Maroon" title="c5">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M9</font></span>)</span> &amp; ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> implies <font color="Maroon" title="c6">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="c5">K</font></span>)</span> ) implies  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:108"/><span class="lab"><font color="Green" title="E94">A1</font></span>: 
<font color="Maroon" title="c3">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M9</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:108"/><span class="lab"><font color="Green" title="E95">A2</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> implies <font color="Maroon" title="c6">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="c5">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">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_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="E1:108_1"/>
(  not <font color="Maroon" title="c4">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="c7">M9</font></span>)</span> or <font color="Maroon" title="c4">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="c7">M9</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_1"/>
 not <font color="Maroon" title="c4">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="c7">M9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:108_1_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T40" target="_self" title="MATRIX13:th.40">Th40</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="108_1_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_2"/><span class="lab"><font color="Green" title="E96">A3</font></span>: 
<font color="Maroon" title="c4">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="c7">M9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">KK</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font>;<br/><span class="kw">set </span><font color="Maroon" title="c9">W</font> =  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M9</font>;<br/><span class="kw">set </span><font color="Maroon" title="c10">Lj</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>);<br/><span class="kw">set </span><font color="Maroon" title="c11">Li</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>);<br/><span class="kw">set </span><font color="Maroon" title="c12">R</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>);<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">Li9</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>), <font color="Maroon" title="c14">Lj9</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>), <font color="Maroon" title="c15">LL</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><a NAME="E3:108_1_2"/><span class="lab"><font color="Green" title="E97">A4</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="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</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="E4:108_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A5</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="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E5:108_1_2"/><span class="lab"><font color="Green" title="E99">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">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="c7">M9</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="E6:108_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A7</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M9</font> <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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A8</font></span>:  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span>) = 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c13">Li9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></span>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c15">LL</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c13">Li9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108_1_2"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></span>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c13">Li9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></span>
<br/>.= 
<font color="Maroon" title="c7">M9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a></span>
;<br/><a NAME="E8:108_1_2"/><span class="lab"><font color="Green" title="E102">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n9</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="E9:108_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A10</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<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#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1_2"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E5:108_1_2"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></span>;<br/><a NAME="E10:108_1_2"/><span class="lab"><font color="Green" title="E104">A11</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="c7">M9</font>,<font color="Maroon" title="c3">j</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="c7">M9</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="E11:108_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A12</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E12:108_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M9</font>,<font color="Maroon" title="c3">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="c7">M9</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="E13:108_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A13</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</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="c7">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E14:108_1_2"/><span class="lab"><font color="Green" title="E107">A14</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T26" title="MATRIX_1:th.26">MATRIX_1:26</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">P1</font>, <font color="Maroon" title="c17">Q1</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="E16:108_1_2"/><span class="lab"><font color="Green" title="E108">A15</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font></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> <font color="Maroon" title="c7">M9</font>
 <span class="kw">and </span><br/><a NAME="E17:108_1_2"/><span class="lab"><font color="Green" title="E109">A16</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">P1</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="c17">Q1</font>
 <span class="kw">and </span><br/><a NAME="E18:108_1_2"/><span class="lab"><font color="Green" title="E110">A17</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">P1</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="c7">M9</font>
 <span class="kw">and </span><br/><a NAME="E19:108_1_2"/><span class="lab"><font color="Green" title="E111">A18</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> <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="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></span>;<br/><a NAME="E20:108_1_2"/><span class="lab"><font color="Green" title="E112">A19</font></span>: 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>) <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="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>;<br/><a NAME="E21:108_1_2"/><span class="lab"><font color="Green" title="E113">A20</font></span>: 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>;<br/><span class="lab"><font color="Green" title="E114">A21</font></span>:  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>) = 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c14">Lj9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:108_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="txt" href="#E6:108_1_2"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></span>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c15">LL</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c14">Lj9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108_1_2"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></span>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c14">Lj9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:108_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></span>
;<br/><div><span class="lab"><font color="Green" title="E115">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_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="E1:108_1_2_3_1"/>
(  not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P1</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_3_1_1"/>
 not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P1</font>
 ; <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> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><a NAME="E2:108_1_2_3_1_1"/><span class="kw">then </span>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="ref" href="matrix13.html#T60" target="_self" title="MATRIX13:th.60">Th60</a></span>;<br/><span class="kw">hence </span><a NAME="E3:108_1_2_3_1_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E18:108_1_2"><span class="lab"><font color="Green" title="E110">A17</font></span></a>, <a class="txt" href="#E19:108_1_2"><span class="lab"><font color="Green" title="E111">A18</font></span></a>, <a class="txt" href="#E21:108_1_2"><span class="lab"><font color="Green" title="E113">A20</font></span></a>, <a class="txt" href="#E20:108_1_2"><span class="lab"><font color="Green" title="E112">A19</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</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="108_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E116">A23</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P1</font>
 ; <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="c7">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c18">SM</font> =  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>);<br/><span class="kw">set </span><font color="Maroon" title="c19">SR</font> =  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>);<br/><a NAME="E2:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E117">A24</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E3:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E118">A25</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c16">P1</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> <font color="Olive" title="b1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></span>;<br/><a NAME="E4:108_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A26</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> <font color="Maroon" title="c16">P1</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">P1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T40" title="FINSEQ_3:th.40">FINSEQ_3:40</a></span>;<br/><a NAME="E5:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E120">A27</font></span>: 
<font color="Maroon" title="c17">Q1</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="c7">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a></span>;<br/><a NAME="E6:108_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A28</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> <font color="Maroon" title="c17">Q1</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T40" title="FINSEQ_3:th.40">FINSEQ_3:40</a></span>;<br/><a NAME="E7:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E122">A29</font></span>: 
 <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> <font color="Maroon" title="c17">Q1</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> <font color="Maroon" title="c7">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E120">A27</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><a NAME="E8:108_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A30</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c17">Q1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c17">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:108_1_2"><span class="lab"><font color="Green" title="E105">A12</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">LjQ</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c17">Q1</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E121">A28</font></span></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/><a NAME="E10:108_1_2_3_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">LjQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T26" title="RELAT_1:th.26">RELAT_1:26</a></span>;<br/><a NAME="E11:108_1_2_3_1_2"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">LjQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E117">A24</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">LjQ</font> = <font color="Maroon" title="c20">LjQ</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c5">K</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E13:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E124">A31</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">LjQ</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="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E8:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E123">A30</font></span></a>, <a class="txt" href="#E6:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E121">A28</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E14:108_1_2_3_1_2"/>
 <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> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E118">A25</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E125">A32</font></span>: 
<font color="Maroon" title="c22">m</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="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c16">P1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E126">A33</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E116">A23</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">m</font> = <font color="Maroon" title="c22">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E125">A32</font></span></a></span>;<br/><a NAME="E19:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E127">A34</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E20:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E128">A35</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</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="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c17">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:108_1_2"><span class="lab"><font color="Green" title="E112">A19</font></span></a>, <a class="txt" href="#E16:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E125">A32</font></span></a>, <a class="txt" href="#E17:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E126">A33</font></span></a>, <a class="txt" href="#E4:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E119">A26</font></span></a>, <a class="txt" href="#E5:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E120">A27</font></span></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></span>;<br/><a NAME="E21:108_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A36</font></span>: 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>)</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="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="txt" href="#E6:108_1_2"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E7:108_1_2"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E21:108_1_2"><span class="lab"><font color="Green" title="E113">A20</font></span></a>, <a class="txt" href="#E17:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E126">A33</font></span></a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></span>;<br/><a NAME="E22:108_1_2_3_1_2"/>
<font color="Maroon" title="c17">Q1</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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a></span>;<br/><a NAME="E23:108_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A37</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</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> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c17">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:108_1_2"><span class="lab"><font color="Green" title="E113">A20</font></span></a>, <a class="txt" href="#E16:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E125">A32</font></span></a>, <a class="txt" href="#E17:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E126">A33</font></span></a>, <a class="txt" href="#E4:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E119">A26</font></span></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></span>;<br/><a NAME="E24:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E131">A38</font></span>: 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c21">LjQ</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:108_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="txt" href="#E6:108_1_2"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E22:108_1_2"><span class="lab"><font color="Green" title="E114">A21</font></span></a>, <a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E21:108_1_2"><span class="lab"><font color="Green" title="E113">A20</font></span></a>, <a class="txt" href="#E17:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E126">A33</font></span></a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></span>;<br/><a NAME="E25:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E132">A39</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c21">LjQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c17">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:108_1_2"><span class="lab"><font color="Green" title="E105">A12</font></span></a>, <a class="txt" href="#E7:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E122">A29</font></span></a>, <a class="ref" href="matrix13.html#T87" target="_self" title="MATRIX13:th.87">Th87</a></span>;<br/><a NAME="E26:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E133">A40</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">LjQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c21">LjQ</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>;<br/><a NAME="E27:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E134">A41</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</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="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><a NAME="E28:108_1_2_3_1_2"/><span class="lab"><font color="Green" title="E135">A42</font></span>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>;<br/> <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>) = 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a></span>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</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="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c21">LjQ</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:108_1_2"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="txt" href="#E4:108_1_2"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="txt" href="#E13:108_1_2"><span class="lab"><font color="Green" title="E106">A13</font></span></a>, <a class="txt" href="#E7:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E122">A29</font></span></a>, <a class="txt" href="#E23:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E130">A37</font></span></a>, <a class="txt" href="#E20:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E128">A35</font></span></a>, <a class="txt" href="#E25:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E132">A39</font></span></a>, <a class="ref" href="matrix13.html#T88" target="_self" title="MATRIX13:th.88">Th88</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A43</font></span>:  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>,<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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>)</span>)</span>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c21">LjQ</font></span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E125">A32</font></span></a>, <a class="txt" href="#E4:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E119">A26</font></span></a>, <a class="txt" href="#E13:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E124">A31</font></span></a>, <a class="txt" href="#E26:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E133">A40</font></span></a>, <a class="txt" href="#E19:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E127">A34</font></span></a>, <a class="txt" href="#E27:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E134">A41</font></span></a>, <a class="ref" href="matrix11.html#T36" title="MATRIX11:th.36">MATRIX11:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>,<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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>)</span>)</span>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p5">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c21">LjQ</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E125">A32</font></span></a>, <a class="txt" href="#E4:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E119">A26</font></span></a>, <a class="txt" href="#E13:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E124">A31</font></span></a>, <a class="ref" href="matrix11.html#T34" title="MATRIX11:th.34">MATRIX11:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E21:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E129">A36</font></span></a>, <a class="txt" href="#E24:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E131">A38</font></span></a>, <a class="txt" href="#E28:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E135">A42</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_2_3"><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="E1:108_1_2_3_1_2_3"/>
(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> <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="c5">K</font> or  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> <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="c5">K</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_2_3_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_3_1_2_3_1"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> <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="c5">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="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:108_1_2_3_1_2_3_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E18:108_1_2"><span class="lab"><font color="Green" title="E110">A17</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</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="108_1_2_3_1_2_3_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_3_1_2_3_2"/><span class="lab"><font color="Green" title="E137">A44</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> <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="c5">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="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c24">j0</font> = <font color="Maroon" title="c3">j</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E94">A1</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_2_3_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="E1:108_1_2_3_1_2_3_2_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P1</font> ) or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp;  not <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P1</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_2_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_3_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E138">A45</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font>
 ; <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> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><span class="kw">then </span> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E136">A43</font></span></a>, <a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D7" title="VECTSP_1:def.7">VECTSP_1:def 7</a></span>
;<br/><a NAME="E3:108_1_2_3_1_2_3_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">a</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="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:108_1_2"><span class="lab"><font color="Green" title="E111">A18</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2_3_2"><span class="lab"><font color="Green" title="E137">A44</font></span></a>, <a class="ref" href="vectsp_1.html#T12" title="VECTSP_1:th.12">VECTSP_1:12</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c6">a</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c5">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_2.html#T2" title="VECTSP_2:th.2">VECTSP_2:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c5">K</font></span>)</span></span>)</span>

<br/>.= 
 <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="c5">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:108_1_2_3_1_2_3_2_1_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E138">A45</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="108_1_2_3_1_2_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_3_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E138">A46</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P1</font> )
 ; <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> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><a NAME="E2:108_1_2_3_1_2_3_2_1_2"/>
 <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> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E118">A25</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:108_1_2_3_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E139">A47</font></span>: 
<font color="Maroon" title="c25">l</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="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c16">P1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:108_1_2_3_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E140">A48</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1_2_3_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E138">A46</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">l</font> = <font color="Maroon" title="c25">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108_1_2_3_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E139">A47</font></span></a></span>;<br/><a NAME="E7:108_1_2_3_1_2_3_2_1_2"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>,<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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c26">l</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E125">A32</font></span></a>, <a class="txt" href="#E17:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E126">A33</font></span></a>, <a class="txt" href="#E4:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E119">A26</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E138">A46</font></span></a>, <a class="txt" href="#E4:108_1_2_3_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E139">A47</font></span></a>, <a class="txt" href="#E5:108_1_2_3_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E140">A48</font></span></a>, <a class="ref" href="matrix11.html#T51" title="MATRIX11:th.51">MATRIX11:51</a></span>;<br/><a NAME="E8:108_1_2_3_1_2_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A49</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</font>,<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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c26">l</font>)</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T12" title="VECTSP_1:th.12">VECTSP_1:12</a></span>;<br/><a NAME="E9:108_1_2_3_1_2_3_2_1_2"/>
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c23">m</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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span>,<font color="Maroon" title="c26">l</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:108_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E20:108_1_2"><span class="lab"><font color="Green" title="E112">A19</font></span></a>, <a class="txt" href="#E17:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E126">A33</font></span></a>, <a class="txt" href="#E4:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E119">A26</font></span></a>, <a class="txt" href="#E5:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E120">A27</font></span></a>, <a class="txt" href="#E24:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E131">A38</font></span></a>, <a class="txt" href="#E28:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E135">A42</font></span></a>, <a class="txt" href="#E4:108_1_2_3_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E139">A47</font></span></a>, <a class="txt" href="#E5:108_1_2_3_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E140">A48</font></span></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></span>;<br/><span class="kw">hence </span><a NAME="E10:108_1_2_3_1_2_3_2_1_2"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:108_1_2"><span class="lab"><font color="Green" title="E111">A18</font></span></a>, <a class="txt" href="#E30:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E136">A43</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2_3_2"><span class="lab"><font color="Green" title="E137">A44</font></span></a>, <a class="txt" href="#E8:108_1_2_3_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E141">A49</font></span></a>, <a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</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="108_1_2_3_1_2_3_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_3_1_2_3_2_1_3"/><span class="lab"><font color="Green" title="E138">A50</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp;  not <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P1</font> )
 ; <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> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c25">Pij</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>;<br/><a NAME="E2:108_1_2_3_1_2_3_2_1_3"/><span class="lab"><font color="Green" title="E139">A51</font></span>: 
 not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P1</font> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/><a NAME="E3:108_1_2_3_1_2_3_2_1_3"/><span class="lab"><font color="Green" title="E140">A52</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"><span class="p3">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">Q1</font></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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E8:108_1_2"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E116">A23</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E138">A50</font></span></a>, <a class="ref" href="matrix13.html#T68" target="_self" title="MATRIX13:th.68">Th68</a></span>;<br/><a NAME="E4:108_1_2_3_1_2_3_2_1_3"/>
<font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span></span>)</span> <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="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:108_1_2"><span class="lab"><font color="Green" title="E111">A18</font></span></a>, <a class="txt" href="#E30:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E136">A43</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2_3_2"><span class="lab"><font color="Green" title="E137">A44</font></span></a>, <a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>;<br/><a NAME="E5:108_1_2_3_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A53</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> <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="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T12" title="VECTSP_1:th.12">VECTSP_1:12</a></span>;<br/><a NAME="E6:108_1_2_3_1_2_3_2_1_3"/>
(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> or  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c16">P1</font>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">Q1</font>)</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E8:108_1_2"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E116">A23</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E138">A50</font></span></a>, <a class="ref" href="matrix13.html#T68" target="_self" title="MATRIX13:th.68">Th68</a></span>;<br/><a NAME="E7:108_1_2_3_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A54</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">Q1</font>)</span>)</span> <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="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E141">A53</font></span></a>, <a class="ref" href="vectsp_2.html#T3" title="VECTSP_2:th.3">VECTSP_2:3</a></span>;<br/><a NAME="E8:108_1_2_3_1_2_3_2_1_3"/>
 not <font color="Maroon" title="c4">i</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="c3">j</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E138">A50</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E9:108_1_2_3_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A55</font></span>: 
 not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E139">A51</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:108_1_2_3_1_2_3_2_1_3"/><span class="lab"><font color="Green" title="E144">A56</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">P1</font> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E8:108_1_2"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E16:108_1_2"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2"><span class="lab"><font color="Green" title="E116">A23</font></span></a>, <a class="txt" href="#E1:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E138">A50</font></span></a>, <a class="ref" href="matrix13.html#T68" target="_self" title="MATRIX13:th.68">Th68</a></span>;<br/><span class="kw">then </span> <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">Q1</font>) = 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c7">M9</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">Q1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
<br/>.= 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">Q1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E3:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E9:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E143">A55</font></span></a>, <a class="ref" href="matrix13.html#T60" target="_self" title="MATRIX13:th.60">Th60</a></span>
<br/>.= 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c24">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">Q1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E10:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E144">A56</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:108_1_2_3_1_2_3_2_1_3"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108_1_2"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E18:108_1_2"><span class="lab"><font color="Green" title="E110">A17</font></span></a>, <a class="txt" href="#E7:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E142">A54</font></span></a>, <a class="txt" href="#E10:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E144">A56</font></span></a>, <a class="txt" href="#E3:108_1_2_3_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</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><span class="kw">consider </span><font color="Maroon" title="c18">P</font>, <font color="Maroon" title="c19">Q</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="E25:108_1_2"/><span class="lab"><font color="Green" title="E116">A57</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font></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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E26:108_1_2"/><span class="lab"><font color="Green" title="E117">A58</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">P</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="c19">Q</font>
 <span class="kw">and </span><br/><a NAME="E27:108_1_2"/><span class="lab"><font color="Green" title="E118">A59</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">P</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E28:108_1_2"/><span class="lab"><font color="Green" title="E119">A60</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span> <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="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></span>;<br/><a NAME="E29:108_1_2"/><span class="lab"><font color="Green" title="E120">A61</font></span>: 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>;<br/><a NAME="E30:108_1_2"/><span class="lab"><font color="Green" title="E121">A62</font></span>: 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>) <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="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_4"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_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="E1:108_1_2_4_1"/>
(  not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_4_1_1"/>
 not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P</font>
 ; <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> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><a NAME="E2:108_1_2_4_1_1"/><span class="kw">then </span>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="ref" href="matrix13.html#T60" target="_self" title="MATRIX13:th.60">Th60</a></span>;<br/><span class="kw">hence </span><a NAME="E3:108_1_2_4_1_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E27:108_1_2"><span class="lab"><font color="Green" title="E118">A59</font></span></a>, <a class="txt" href="#E28:108_1_2"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="txt" href="#E29:108_1_2"><span class="lab"><font color="Green" title="E120">A61</font></span></a>, <a class="txt" href="#E30:108_1_2"><span class="lab"><font color="Green" title="E121">A62</font></span></a>, <a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</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="108_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E122">A63</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P</font>
 ; <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> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c20">SM</font> =  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>);<br/><span class="kw">set </span><font color="Maroon" title="c21">SR</font> =  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>);<br/><a NAME="E2:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E123">A64</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E3:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E124">A65</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c18">P</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> <font color="Olive" title="b1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></span>;<br/><a NAME="E4:108_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A66</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> <font color="Maroon" title="c18">P</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T40" title="FINSEQ_3:th.40">FINSEQ_3:40</a></span>;<br/><a NAME="E5:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E126">A67</font></span>: 
<font color="Maroon" title="c19">Q</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="c7">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a></span>;<br/><a NAME="E6:108_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A68</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> <font color="Maroon" title="c19">Q</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c19">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T40" title="FINSEQ_3:th.40">FINSEQ_3:40</a></span>;<br/><a NAME="E7:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E128">A69</font></span>: 
 <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> <font color="Maroon" title="c19">Q</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> <font color="Maroon" title="c7">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E126">A67</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><a NAME="E8:108_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A70</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c19">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c19">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:108_1_2"><span class="lab"><font color="Green" title="E105">A12</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">LjQ</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c19">Q</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E127">A68</font></span></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/><a NAME="E10:108_1_2_4_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">LjQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T26" title="RELAT_1:th.26">RELAT_1:26</a></span>;<br/><a NAME="E11:108_1_2_4_1_2"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">LjQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E123">A64</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">LjQ</font> = <font color="Maroon" title="c22">LjQ</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c5">K</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E13:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E130">A71</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">LjQ</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="c18">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E8:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E129">A70</font></span></a>, <a class="txt" href="#E6:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E127">A68</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E14:108_1_2_4_1_2"/>
 <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> <font color="Maroon" title="c18">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E124">A65</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E131">A72</font></span>: 
<font color="Maroon" title="c24">m</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="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c18">P</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E132">A73</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c18">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">m</font> = <font color="Maroon" title="c24">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E131">A72</font></span></a></span>;<br/><a NAME="E19:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E133">A74</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E20:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E134">A75</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</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="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c19">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:108_1_2"><span class="lab"><font color="Green" title="E121">A62</font></span></a>, <a class="txt" href="#E16:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E131">A72</font></span></a>, <a class="txt" href="#E17:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E132">A73</font></span></a>, <a class="txt" href="#E4:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E125">A66</font></span></a>, <a class="txt" href="#E5:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E126">A67</font></span></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></span>;<br/><a NAME="E21:108_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A76</font></span>: 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>)</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="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="txt" href="#E6:108_1_2"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E7:108_1_2"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E29:108_1_2"><span class="lab"><font color="Green" title="E120">A61</font></span></a>, <a class="txt" href="#E17:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E132">A73</font></span></a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></span>;<br/><a NAME="E22:108_1_2_4_1_2"/>
<font color="Maroon" title="c19">Q</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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a></span>;<br/><a NAME="E23:108_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A77</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</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> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c19">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:108_1_2"><span class="lab"><font color="Green" title="E120">A61</font></span></a>, <a class="txt" href="#E16:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E131">A72</font></span></a>, <a class="txt" href="#E17:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E132">A73</font></span></a>, <a class="txt" href="#E4:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E125">A66</font></span></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></span>;<br/><a NAME="E24:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E137">A78</font></span>: 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>,<font color="Maroon" title="c23">LjQ</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:108_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="txt" href="#E6:108_1_2"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E22:108_1_2"><span class="lab"><font color="Green" title="E114">A21</font></span></a>, <a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E29:108_1_2"><span class="lab"><font color="Green" title="E120">A61</font></span></a>, <a class="txt" href="#E17:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E132">A73</font></span></a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></span>;<br/><a NAME="E25:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E138">A79</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c23">LjQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c19">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:108_1_2"><span class="lab"><font color="Green" title="E105">A12</font></span></a>, <a class="txt" href="#E7:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E128">A69</font></span></a>, <a class="ref" href="matrix13.html#T87" target="_self" title="MATRIX13:th.87">Th87</a></span>;<br/><a NAME="E26:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E139">A80</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">LjQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c23">LjQ</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>;<br/><a NAME="E27:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E140">A81</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</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="c18">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><a NAME="E28:108_1_2_4_1_2"/><span class="lab"><font color="Green" title="E141">A82</font></span>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>;<br/> <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>) = 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a></span>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</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="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c23">LjQ</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:108_1_2"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="txt" href="#E4:108_1_2"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="txt" href="#E13:108_1_2"><span class="lab"><font color="Green" title="E106">A13</font></span></a>, <a class="txt" href="#E7:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E128">A69</font></span></a>, <a class="txt" href="#E23:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E136">A77</font></span></a>, <a class="txt" href="#E20:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E134">A75</font></span></a>, <a class="txt" href="#E25:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E138">A79</font></span></a>, <a class="ref" href="matrix13.html#T88" target="_self" title="MATRIX13:th.88">Th88</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A83</font></span>:  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>,<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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>)</span>)</span>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c23">LjQ</font></span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E131">A72</font></span></a>, <a class="txt" href="#E4:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E125">A66</font></span></a>, <a class="txt" href="#E13:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E130">A71</font></span></a>, <a class="txt" href="#E26:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E139">A80</font></span></a>, <a class="txt" href="#E19:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E133">A74</font></span></a>, <a class="txt" href="#E27:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E140">A81</font></span></a>, <a class="ref" href="matrix11.html#T36" title="MATRIX11:th.36">MATRIX11:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>,<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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>)</span>)</span>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p5">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>,<font color="Maroon" title="c23">LjQ</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E131">A72</font></span></a>, <a class="txt" href="#E4:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E125">A66</font></span></a>, <a class="txt" href="#E13:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E130">A71</font></span></a>, <a class="ref" href="matrix11.html#T34" title="MATRIX11:th.34">MATRIX11:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E21:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E135">A76</font></span></a>, <a class="txt" href="#E24:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E137">A78</font></span></a>, <a class="txt" href="#E28:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E141">A82</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_2_3"><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="E1:108_1_2_4_1_2_3"/>
(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span> <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="c5">K</font> or  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span> <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="c5">K</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_2_3_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_4_1_2_3_1"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span> <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="c5">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="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:108_1_2_4_1_2_3_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E27:108_1_2"><span class="lab"><font color="Green" title="E118">A59</font></span></a>, <a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</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="108_1_2_4_1_2_3_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_4_1_2_3_2"/><span class="lab"><font color="Green" title="E143">A84</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span> <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="c5">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="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c26">j0</font> = <font color="Maroon" title="c3">j</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E94">A1</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_2_3_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="E1:108_1_2_4_1_2_3_2_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P</font> ) or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp;  not <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_2_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_4_1_2_3_2_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font>
 ; <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> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><span class="kw">then </span> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E142">A83</font></span></a>, <a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D7" title="VECTSP_1:def.7">VECTSP_1:def 7</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:108_1_2_4_1_2_3_2_1_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:108_1_2"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="txt" href="#E1:108_1_2_4_1_2_3_2"><span class="lab"><font color="Green" title="E143">A84</font></span></a>, <a class="ref" href="vectsp_1.html#T12" title="VECTSP_1:th.12">VECTSP_1:12</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="108_1_2_4_1_2_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_4_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E144">A85</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P</font> )
 ; <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> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><a NAME="E2:108_1_2_4_1_2_3_2_1_2"/>
 <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> <font color="Maroon" title="c18">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E124">A65</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:108_1_2_4_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E145">A86</font></span>: 
<font color="Maroon" title="c27">l</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="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c18">P</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:108_1_2_4_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E146">A87</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c18">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1_2_4_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E144">A85</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c28">l</font> = <font color="Maroon" title="c27">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108_1_2_4_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E145">A86</font></span></a></span>;<br/><a NAME="E7:108_1_2_4_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E147">A88</font></span>: 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c28">l</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:108_1_2"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E30:108_1_2"><span class="lab"><font color="Green" title="E121">A62</font></span></a>, <a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E17:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E132">A73</font></span></a>, <a class="txt" href="#E4:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E125">A66</font></span></a>, <a class="txt" href="#E5:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E126">A67</font></span></a>, <a class="txt" href="#E24:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E137">A78</font></span></a>, <a class="txt" href="#E28:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E141">A82</font></span></a>, <a class="txt" href="#E4:108_1_2_4_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E145">A86</font></span></a>, <a class="txt" href="#E5:108_1_2_4_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E146">A87</font></span></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></span>;<br/><a NAME="E8:108_1_2_4_1_2_3_2_1_2"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>,<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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c28">l</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E131">A72</font></span></a>, <a class="txt" href="#E17:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E132">A73</font></span></a>, <a class="txt" href="#E4:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E125">A66</font></span></a>, <a class="txt" href="#E1:108_1_2_4_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E144">A85</font></span></a>, <a class="txt" href="#E4:108_1_2_4_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E145">A86</font></span></a>, <a class="txt" href="#E5:108_1_2_4_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E146">A87</font></span></a>, <a class="ref" href="matrix11.html#T51" title="MATRIX11:th.51">MATRIX11:51</a></span>;<br/><span class="kw">then </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font> = 
<font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c25">m</font>,<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"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span>,<font color="Maroon" title="c28">l</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T12" title="VECTSP_1:th.12">VECTSP_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1_2_4_1_2_3_2"><span class="lab"><font color="Green" title="E143">A84</font></span></a>, <a class="txt" href="#E7:108_1_2_4_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E147">A88</font></span></a>, <a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:108_1_2_4_1_2_3_2_1_2"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:108_1_2"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="txt" href="#E30:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E142">A83</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="108_1_2_4_1_2_3_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_4_1_2_3_2_1_3"/><span class="lab"><font color="Green" title="E144">A89</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp;  not <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P</font> )
 ; <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> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c27">Pij</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">P</font> <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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c26">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>;<br/><a NAME="E2:108_1_2_4_1_2_3_2_1_3"/><span class="lab"><font color="Green" title="E145">A90</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">P</font> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c18">P</font> <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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c26">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E8:108_1_2"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E1:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="txt" href="#E1:108_1_2_4_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E144">A89</font></span></a>, <a class="ref" href="matrix13.html#T68" target="_self" title="MATRIX13:th.68">Th68</a></span>;<br/><a NAME="E3:108_1_2_4_1_2_3_2_1_3"/>
<font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span> <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="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:108_1_2"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="txt" href="#E30:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E142">A83</font></span></a>, <a class="txt" href="#E1:108_1_2_4_1_2_3_2"><span class="lab"><font color="Green" title="E143">A84</font></span></a>, <a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>;<br/><a NAME="E4:108_1_2_4_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A91</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span> <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="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T12" title="VECTSP_1:th.12">VECTSP_1:12</a></span>;<br/><a NAME="E5:108_1_2_4_1_2_3_2_1_3"/>
(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c26">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">Q</font>)</span>)</span> or  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">Q</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c18">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c26">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">Q</font>)</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E8:108_1_2"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E1:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="txt" href="#E1:108_1_2_4_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E144">A89</font></span></a>, <a class="ref" href="matrix13.html#T68" target="_self" title="MATRIX13:th.68">Th68</a></span>;<br/><a NAME="E6:108_1_2_4_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A92</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c7">M9</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c26">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">Q</font>)</span>)</span> <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="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108_1_2_4_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E146">A91</font></span></a>, <a class="ref" href="vectsp_2.html#T3" title="VECTSP_2:th.3">VECTSP_2:3</a></span>;<br/><a NAME="E7:108_1_2_4_1_2_3_2_1_3"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c26">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">Q</font></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> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E8:108_1_2"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E14:108_1_2"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E1:108_1_2_4_1_2"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="txt" href="#E1:108_1_2_4_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E144">A89</font></span></a>, <a class="ref" href="matrix13.html#T68" target="_self" title="MATRIX13:th.68">Th68</a></span>;<br/><span class="kw">hence </span><a NAME="E8:108_1_2_4_1_2_3_2_1_3"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <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> <font color="Maroon" title="c7">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:108_1_2"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="txt" href="#E27:108_1_2"><span class="lab"><font color="Green" title="E118">A59</font></span></a>, <a class="txt" href="#E6:108_1_2_4_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E147">A92</font></span></a>, <a class="txt" href="#E2:108_1_2_4_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E145">A90</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</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><span class="kw">hence </span><a NAME="E32:108_1_2"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M9</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c7">M9</font>,<font color="Maroon" title="c4">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="c7">M9</font>,<font color="Maroon" title="c4">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="c6">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="c7">M9</font>,<font color="Maroon" title="c3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:108_1_2"><span class="lab"><font color="Green" title="E115">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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></div><span class="kw">end;</span></div>

</div>
