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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</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">A9</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">K</font><br/>  for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; ( <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j</font> implies <font color="Olive" title="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="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Olive" title="b3">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Olive" title="b3">A9</font>,<font color="Olive" title="b4">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">A9</font>,<font color="Olive" title="b4">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">A9</font>,<font color="Olive" title="b5">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="c3">K</font><br/>  for <font color="Olive" title="b2">A9</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">K</font><br/>  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; ( <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j</font> implies <font color="Olive" title="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="c3">K</font></span>)</span> ) holds <br/> <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Olive" title="b2">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Olive" title="b2">A9</font>,<font color="Olive" title="b3">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">A9</font>,<font color="Olive" title="b3">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">A9</font>,<font color="Olive" title="b4">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">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">A9</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">K</font><br/>  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; ( <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">j</font> implies <font color="Maroon" title="c4">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="c3">K</font></span>)</span> ) holds <br/> <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Olive" title="b1">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Olive" title="b1">A9</font>,<font color="Olive" title="b2">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">A9</font>,<font color="Olive" title="b2">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="c4">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">A9</font>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">A9</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> implies <font color="Maroon" title="c4">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="c3">K</font></span>)</span> ) holds <br/> <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c5">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">A9</font>,<font color="Olive" title="b1">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="c5">A9</font>,<font color="Olive" title="b1">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="c4">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="c5">A9</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> implies <font color="Maroon" title="c4">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="c3">K</font></span>)</span> ) implies  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c5">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">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="c5">A9</font>,<font color="Maroon" title="c6">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="c4">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="c5">A9</font>,<font color="Maroon" title="c7">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="E1:80"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
<font color="Maroon" title="c7">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> <font color="Maroon" title="c1">m</font>
 <span class="kw">and </span><br/><a NAME="E2:80"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:80"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> implies <font color="Maroon" title="c4">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="c3">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="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c5">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">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="c5">A9</font>,<font color="Maroon" title="c6">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="c4">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="c5">A9</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">L</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A9</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">R</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">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="c5">A9</font>,<font color="Maroon" title="c6">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="c4">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="c5">A9</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>);<br/>
<a NAME="E4:80"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/>
<a NAME="E5:80"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A5</font></span>: 
 <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="c5">A9</font>,<font color="Maroon" title="c6">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="c5">A9</font>,<font color="Maroon" title="c6">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="c4">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="c5">A9</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<a NAME="E6:80"/>
 <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A9</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">C</font> =  <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A9</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">m</font>,1,<font color="Maroon" title="c3">K</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/>
<span class="kw">set </span><font color="Maroon" title="c11">RC</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>);<br/>
<a NAME="E8:80"/><span class="lab"><font color="Green" title="E67">A6</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A9</font></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T32" target="_self" title="MATRIX15:th.32">Th32</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">i9</font>, <font color="Maroon" title="c13">j9</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c13">j9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c10">C</font> implies <font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c13">j9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c13">j9</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:80_1"/><span class="lab"><font color="Green" title="E68">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c13">j9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c10">C</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="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c13">j9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c13">j9</font>)</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c14">I</font> = <font color="Maroon" title="c12">i9</font>, <font color="Maroon" title="c15">J</font> = <font color="Maroon" title="c13">j9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E3:80_1"/><span class="lab"><font color="Green" title="E69">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c10">C</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_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="E1:80_1_1_1"/>
( <font color="Maroon" title="c12">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> or <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">i9</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_1_1"/><span class="lab"><font color="Green" title="E70">A9</font></span>: 
<font color="Maroon" title="c12">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</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="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">I</font>,<font color="Maroon" title="c15">J</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">I</font>,<font color="Maroon" title="c15">J</font>)</span><br/><div class="add"><a NAME="E2:80_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A10</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c10">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E3:80_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A11</font></span>: 
<font color="Maroon" title="c13">j9</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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E4:80_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c13">j9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_1_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E5:80_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c13">j9</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_1_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="txt" href="#E3:80_1_1_1_1"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="fvsum_1.html#T51" title="FVSUM_1:th.51">FVSUM_1:51</a></span>;<br/><a NAME="E6:80_1_1_1_1"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E7:80_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c13">j9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c10">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E3:80_1_1_1_1"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E8:80_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c13">j9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_1_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="txt" href="#E3:80_1_1_1_1"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j9</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c13">j9</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c13">j9</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_1_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="txt" href="#E3:80_1_1_1_1"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="txt" href="#E5:80_1_1_1_1"><span class="lab"><font color="Green" title="E73">A12</font></span></a>, <a class="ref" href="fvsum_1.html#T18" title="FVSUM_1:th.18">FVSUM_1:18</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="c3">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c13">j9</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:80_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E1:80_1_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="matrix_3.html#T1" title="MATRIX_3:th.1">MATRIX_3:1</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="c3">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E7:80_1_1_1_1"><span class="lab"><font color="Green" title="E74">A13</font></span></a>, <a class="ref" href="matrix_3.html#T1" title="MATRIX_3:th.1">MATRIX_3:1</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="c3">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T6" title="VECTSP_1:th.6">VECTSP_1:6</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font>
<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/>.= 
<font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c13">j9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:80_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="ref" href="matrix_3.html#T1" title="MATRIX_3:th.1">MATRIX_3:1</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:80_1_1_1_1"/>
<font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">I</font>,<font color="Maroon" title="c15">J</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">I</font>,<font color="Maroon" title="c15">J</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E69">A8</font></span></a>, <a class="txt" href="#E1:80_1_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</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="80_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_1_2"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">i9</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="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">I</font>,<font color="Maroon" title="c15">J</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">I</font>,<font color="Maroon" title="c15">J</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:80_1_1_1_2"/>
<font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">I</font>,<font color="Maroon" title="c15">J</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">I</font>,<font color="Maroon" title="c15">J</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E69">A8</font></span></a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</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><span class="kw">hence </span><a NAME="E5:80_1"/>
<font color="Maroon" title="c10">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c13">j9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">i9</font>,<font color="Maroon" title="c13">j9</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:80"/><span class="kw">then </span>
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c6">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="c10">C</font>,<font color="Maroon" title="c6">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="c4">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></span>;<br/>
<a NAME="E11:80"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A14</font></span>: 
 <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c10">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">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="c5">A9</font>,<font color="Maroon" title="c6">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="c4">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="c5">A9</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c10">C</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="matrix15.html#T40" target="_self" title="MATRIX15:th.40">Th40</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">SR</font> =  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">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="c5">A9</font>,<font color="Maroon" title="c6">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="c4">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="c5">A9</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>;<br/>
<a NAME="E12:80"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">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="c5">A9</font>,<font color="Maroon" title="c6">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="c4">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="c5">A9</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<a NAME="E13:80"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A15</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">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="c5">A9</font>,<font color="Maroon" title="c6">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="c4">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="c5">A9</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">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="c5">A9</font>,<font color="Maroon" title="c6">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="c4">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="c5">A9</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A9</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="txt" href="#E5:80"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="ref" href="matrix15.html#D5" target="_self" title="MATRIX15:def.5">Def5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">SA</font> =  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c5">A9</font>;<br/>
<a NAME="E14:80"/><span class="lab"><font color="Green" title="E70">A16</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<a NAME="E15:80"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c5">A9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c5">A9</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A9</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="matrix15.html#D5" target="_self" title="MATRIX15:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:80"/>
 <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c5">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">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="c5">A9</font>,<font color="Maroon" title="c6">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="c4">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="c5">A9</font>,<font color="Maroon" title="c7">j</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E70">A16</font></span></a>, <a class="txt" href="#E5:80"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="txt" href="#E11:80"><span class="lab"><font color="Green" title="E68">A14</font></span></a>, <a class="txt" href="#E13:80"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</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/>


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