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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</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="Red">::  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="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">N</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b2">A</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">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="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">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>  holds <br/> ex <font color="Olive" title="b4">MVectors</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">K</font> st <br/>(  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b4">MVectors</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b3">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Olive" title="b1">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b4">MVectors</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Olive" title="b3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b2">A</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b3">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Olive" title="b4">MVectors</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Olive" title="b2">A</font> )</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font><br/> for <font color="Olive" title="b2">N</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">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="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">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>  holds <br/> ex <font color="Olive" title="b3">MVectors</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font> st <br/>(  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b3">MVectors</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b3">MVectors</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Olive" title="b2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b1">A</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Olive" title="b3">MVectors</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">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="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">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>  holds <br/> ex <font color="Olive" title="b2">MVectors</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font> st <br/>(  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b2">MVectors</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b2">MVectors</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Olive" title="b2">MVectors</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c4">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">N</font> be   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">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="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">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>  implies  ex <font color="Olive" title="b1">MVectors</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font> st <br/>(  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b1">MVectors</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b1">MVectors</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Olive" title="b1">MVectors</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c4">A</font> ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83"/><span class="lab"><font color="Green" title="E65">A1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E2:83"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
<font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">and </span><br/><a NAME="E3:83"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E4:83"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c1">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="E5:83"/><span class="lab"><font color="Green" title="E69">A5</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">MVectors</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font> st <br/>(  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b1">MVectors</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b1">MVectors</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Olive" title="b1">MVectors</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c4">A</font> )</span><br/>

<a NAME="E6:83"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></span>;<br/>
<a NAME="E7:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A6</font></span>: 
<font color="Maroon" title="c2">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> 
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">MV</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font><span class="kw"> such that </span><br/><a NAME="E9:83"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">MV</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:83"/><span class="lab"><font color="Green" title="E72">A8</font></span>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">MV</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:83"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
for <font color="Olive" title="b1">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">M</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">l</font>,<font color="Maroon" title="c3">K</font>  st ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Olive" title="b3">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> <font color="Olive" title="b1">l</font> or  ex <font color="Olive" title="b4">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Olive" title="b4">j</font> ) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span> ) ) holds <br/><font color="Olive" title="b2">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="matrix15.html#T49" target="_self" title="MATRIX15:th.49">Th49</a></span>;<br/>
<a NAME="E12:83"/><span class="lab"><font color="Green" title="E74">A10</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">MV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E13:83"/><span class="lab"><font color="Green" title="E75">A11</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">MV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">MV</font> <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="c6">MV</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T74" title="MATRIX13:th.74">MATRIX13:74</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></span>;<br/>
<a NAME="E14:83"/><span class="lab"><font color="Green" title="E76">A12</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">MV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></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="#E5:83"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E15:83"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E16:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A13</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">MV</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></span>;<br/>
<a NAME="E17:83"/><span class="lab"><font color="Green" title="E78">A14</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E18:83"/><span class="lab"><font color="Green" title="E79">A15</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E19:83"/>
 <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> <font color="Maroon" title="c4">A</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="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> <font color="Maroon" title="c4">A</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="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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><br/>

<span class="kw">assume </span><a NAME="E1:83_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> <font color="Maroon" title="c4">A</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="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:83_1"/><span class="lab"><font color="Green" title="E80">A16</font></span>: 
<font color="Maroon" title="c8">k</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"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:83_1"/><span class="lab"><font color="Green" title="E81">A17</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T103" title="MATRIX13:th.103">MATRIX13:103</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">C</font> =  <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E82">A18</font></span>: <font color="Maroon" title="c2">m</font> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">MV</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>
<br/>.= 
 <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="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">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> 1 or  ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Olive" title="b1">j</font> ) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:83_1_2"/>
<font color="Maroon" title="c10">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> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Olive" title="b1">j</font> ) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span> )</span><br/><a NAME="E2:83_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A19</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:83_1_2"/>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="txt" href="#E5:83_1"><span class="lab"><font color="Green" title="E82">A18</font></span></a>, <a class="ref" href="matrix15.html#T26" target="_self" title="MATRIX15:th.26">Th26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:83_1_2"/>
(  ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Olive" title="b1">j</font> ) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E80">A16</font></span></a>, <a class="txt" href="#E2:83_1_2"><span class="lab"><font color="Green" title="E83">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:83_1"/><span class="kw">then </span>
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:83"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="txt" href="#E5:83_1"><span class="lab"><font color="Green" title="E82">A18</font></span></a></span>;<br/>
<a NAME="E8:83_1"/><span class="kw">then </span>
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:83"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="ref" href="matrix15.html#T32" target="_self" title="MATRIX15:th.32">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:83_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> <font color="Maroon" title="c4">A</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="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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="#E4:83_1"><span class="lab"><font color="Green" title="E81">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:83"/><span class="kw">then </span>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> <font color="Maroon" title="c4">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="ref" href="vectsp_7.html#T18" title="VECTSP_7:th.18">VECTSP_7:18</a></span>;<br/>
<a NAME="E21:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A20</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="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span></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 <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> <font color="Maroon" title="c4">A</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p4">(<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>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></span>;<br/>
<a NAME="E22:83"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <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">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E69">A5</font></span></a></span>;<br/>
<a NAME="E23:83"/><span class="kw">then </span>
<font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></span>;<br/>
<a NAME="E24:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A21</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_7.html#T16" title="MATRIX_7:th.16">MATRIX_7:16</a></span>;<br/>
<a NAME="E25:83"/><span class="lab"><font color="Green" title="E82">A22</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
 
;<br/>
<a NAME="E26:83"/><span class="lab"><font color="Green" title="E83">A23</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></span>;<br/>
<a NAME="E27:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A24</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E28:83"/><span class="lab"><font color="Green" title="E85">A25</font></span>: 
 <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"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E26:83"><span class="lab"><font color="Green" title="E83">A23</font></span></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></span>;<br/>
<a NAME="E29:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A26</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></span>;<br/>
<a NAME="E30:83"/>
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">MV</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a>, <a class="ref" href="matrix13.html#D3" title="MATRIX13:def.3">MATRIX13:def 3</a></span>;<br/>
<a NAME="E31:83"/><span class="kw">then </span>
<font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">MV</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E29:83"><span class="lab"><font color="Green" title="E86">A26</font></span></a>, <a class="txt" href="#E16:83"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="txt" href="#E24:83"><span class="lab"><font color="Green" title="E81">A21</font></span></a>, <a class="txt" href="#E25:83"><span class="lab"><font color="Green" title="E82">A22</font></span></a>, <a class="ref" href="matrix13.html#D4" title="MATRIX13:def.4">MATRIX13:def 4</a></span>;<br/>
<a NAME="E32:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A27</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">MV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:83"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E33:83"/><span class="lab"><font color="Green" title="E88">A28</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> <font color="Maroon" title="c4">A</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 <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c7">SN</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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="c4">A</font></span>)</span> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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="c4">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:83_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> <font color="Maroon" title="c4">A</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="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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="#E7:83"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="ref" href="matrix15.html#D5" target="_self" title="MATRIX15:def.5">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c3">K</font><span class="kw"> such that </span><br/><a NAME="E4:83_2"/><span class="lab"><font color="Green" title="E89">A29</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">and </span><br/><a NAME="E5:83_2"/><span class="lab"><font color="Green" title="E90">A30</font></span>: 
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<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>)</span>
 ;<br/>
<a NAME="E6:83_2"/><span class="lab"><font color="Green" title="E91">A31</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E5:83_2"><span class="lab"><font color="Green" title="E90">A30</font></span></a>, <a class="ref" href="matrix15.html#T59" target="_self" title="MATRIX15:th.59">Th59</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">MV</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span>)</span> <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="c6">MV</font>,$1</span>)</span>;<br/>
<a NAME="E7:83_2"/><span class="lab"><font color="Green" title="E92">A32</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</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/>
<span class="kw">consider </span><font color="Maroon" title="c10">M</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">MV</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font><span class="kw"> such that </span><br/><a NAME="E9:83_2"/><span class="lab"><font color="Green" title="E93">A33</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E10:83_2"/><span class="lab"><font color="Green" title="E94">A34</font></span>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">M</font> holds <br/><font color="Maroon" title="c10">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span>
<a NAME="E11:83_2"/><span class="lab"><font color="Green" title="E95">A35</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_2"><span class="lab"><font color="Green" title="E93">A33</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">M</font> = <font color="Maroon" title="c10">M</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">M1</font> =  <a href="matrix13.html#K11" title="MATRIX13:func.11">FinS2MX</a> <font color="Maroon" title="c11">M</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_2"><span class="lab"><font color="Green" title="E93">A33</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> implies  ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> st  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="c6">MV</font>,<font color="Maroon" title="c13">i</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:83_2_1"/><span class="lab"><font color="Green" title="E96">A36</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> st  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="c6">MV</font>,<font color="Maroon" title="c13">i</font></span>)</span></span><br/><span class="kw">take </span><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c9">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">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="c6">MV</font>,<font color="Maroon" title="c13">i</font></span>)</span></span><br/><span class="kw">thus </span> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c13">i</font> = 
<font color="Maroon" title="c12">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_1"><span class="lab"><font color="Green" title="E96">A36</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>
<br/>.= 
<font color="Maroon" title="c14">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="c6">MV</font>,<font color="Maroon" title="c13">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_2"><span class="lab"><font color="Green" title="E94">A34</font></span></a>, <a class="txt" href="#E11:83_2"><span class="lab"><font color="Green" title="E95">A35</font></span></a>, <a class="txt" href="#E1:83_2_1"><span class="lab"><font color="Green" title="E96">A36</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">L</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font><span class="kw"> such that </span><br/><a NAME="E16:83_2"/><span class="lab"><font color="Green" title="E96">A37</font></span>: 
<font color="Maroon" title="c13">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p1">(<span class="default"><a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <font color="Maroon" title="c6">MV</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">M1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:83"><span class="lab"><font color="Green" title="E87">A27</font></span></a>, <a class="ref" href="matrix13.html#T105" title="MATRIX13:th.105">MATRIX13:105</a>, <a class="ref" href="matrix13.html#T108" title="MATRIX13:th.108">MATRIX13:108</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">SumL</font> =  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c13">L</font>, <font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c9">f</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E91">A31</font></span></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> implies <font color="Maroon" title="c14">SumL</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:83_2_2"/><span class="lab"><font color="Green" title="E97">A38</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">SumL</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span><br/><a NAME="E2:83_2_2"/><span class="lab"><font color="Green" title="E98">A39</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><div><span class="lab"><font color="Green" title="E99">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_2_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="E1:83_2_2_1_1"/>
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> or  not <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E100">A41</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c17">X</font> being   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c3">K</font><span class="kw"> such that </span><br/><a NAME="E3:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E101">A42</font></span>: 
<font color="Maroon" title="c17">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c15">f</font>
 <span class="kw">and </span><br/><a NAME="E4:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E102">A43</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E5:83_2_2_1_1_1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c17">X</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="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<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>)</span>
 <span class="kw">and </span><br/><a NAME="E6:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E103">A44</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c17">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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="#E5:83_2"><span class="lab"><font color="Green" title="E90">A30</font></span></a></span>;<br/><a NAME="E7:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E104">A45</font></span>: 
<font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c17">X</font>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E3:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E101">A42</font></span></a>, <a class="txt" href="#E4:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E102">A43</font></span></a>, <a class="ref" href="matrix15.html#T26" target="_self" title="MATRIX15:th.26">Th26</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">F</font> = <font color="Maroon" title="c15">f</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><a NAME="E9:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E105">A46</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> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><a NAME="E10:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E106">A47</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="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</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="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E12:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E107">A48</font></span>: 
<font color="Maroon" title="c19">x</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="c5">N</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E108">A49</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E100">A41</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</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="#E12:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E107">A48</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font>;<br/><a NAME="E15:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E109">A50</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span>,<font color="Maroon" title="c18">F</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> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">mN</font>, <font color="Maroon" title="c23">mSN</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c3">K</font><span class="kw"> such that </span><br/><a NAME="E17:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E110">A51</font></span>: 
<font color="Maroon" title="c22">mN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span>,<font color="Maroon" title="c18">F</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="c5">N</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E18:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E111">A52</font></span>: 
<font color="Maroon" title="c23">mSN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span>,<font color="Maroon" title="c18">F</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E19:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E112">A53</font></span>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span>,<font color="Maroon" title="c18">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c22">mN</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c23">mSN</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="matrix15.html#E78"><span class="lab"><font color="Green" title="E63">Lm7</font></span></a></span>;<br/><a NAME="E20:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E113">A54</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="c5">N</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="c5">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a></span>;<br/><a NAME="E21:83_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A55</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">mN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E10:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E106">A47</font></span></a>, <a class="txt" href="#E15:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E109">A50</font></span></a>, <a class="txt" href="#E17:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E110">A51</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/><a NAME="E22:83_2_2_1_1_1"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></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="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/><a NAME="E23:83_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A56</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c20">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E20:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E113">A54</font></span></a>, <a class="txt" href="#E12:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E107">A48</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E24:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E116">A57</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E106">A47</font></span></a>, <a class="txt" href="#E12:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E107">A48</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E25:83_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A58</font></span>: 
( <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E91">A31</font></span></a>, <a class="txt" href="#E7:83_2"><span class="lab"><font color="Green" title="E92">A32</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="lab"><font color="Green" title="E118">A59</font></span>: <font color="Maroon" title="c20">x</font> = 
<span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E20:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E113">A54</font></span></a>, <a class="txt" href="#E12:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E107">A48</font></span></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_2_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">mN</font> &amp; <font color="Maroon" title="c24">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">x</font> implies <font color="Maroon" title="c22">mN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83_2_2_1_1_1_2"/><span class="lab"><font color="Green" title="E119">A60</font></span>: 
<font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">mN</font>
 <span class="kw">and </span><br/><a NAME="E2:83_2_2_1_1_1_2"/><span class="lab"><font color="Green" title="E120">A61</font></span>: 
<font color="Maroon" title="c24">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">mN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span><br/><a NAME="E3:83_2_2_1_1_1_2"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></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="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/><a NAME="E4:83_2_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A62</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c24">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E20:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E113">A54</font></span></a>, <a class="txt" href="#E12:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E107">A48</font></span></a>, <a class="txt" href="#E21:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E114">A55</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E5:83_2_2_1_1_1_2"/><span class="lab"><font color="Green" title="E122">A63</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E10:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E106">A47</font></span></a>, <a class="txt" href="#E20:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E113">A54</font></span></a>, <a class="txt" href="#E21:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E114">A55</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E6:83_2_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A64</font></span>: 
( <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E91">A31</font></span></a>, <a class="txt" href="#E7:83_2"><span class="lab"><font color="Green" title="E92">A32</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c22">mN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> = 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span>,<font color="Maroon" title="c18">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E110">A51</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E26:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E118">A59</font></span></a>, <a class="txt" href="#E5:83_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="txt" href="#E6:83_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E123">A64</font></span></a>, <a class="ref" href="fvsum_1.html#T74" title="FVSUM_1:th.74">FVSUM_1:74</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c24">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E4:83_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E121">A62</font></span></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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="txt" href="#E2:83_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E120">A61</font></span></a>, <a class="txt" href="#E4:83_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E121">A62</font></span></a>, <a class="ref" href="matrix_1.html#D12" title="MATRIX_1:def.12">MATRIX_1:def 12</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="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E28:83_2_2_1_1_1"/><span class="kw">then </span>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c22">mN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">mN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E20:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E113">A54</font></span></a>, <a class="txt" href="#E12:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E107">A48</font></span></a>, <a class="txt" href="#E21:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E114">A55</font></span></a>, <a class="ref" href="matrix_3.html#T14" title="MATRIX_3:th.14">MATRIX_3:14</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A65</font></span>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c22">mN</font> = 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span>,<font color="Maroon" title="c18">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E20:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E113">A54</font></span></a>, <a class="txt" href="#E12:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E107">A48</font></span></a>, <a class="txt" href="#E17:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E110">A51</font></span></a>, <a class="txt" href="#E21:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E114">A55</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E26:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E118">A59</font></span></a>, <a class="txt" href="#E24:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E25:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="ref" href="fvsum_1.html#T74" title="FVSUM_1:th.74">FVSUM_1:74</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c20">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E23:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E115">A56</font></span></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E115">A56</font></span></a>, <a class="ref" href="matrix_1.html#D12" title="MATRIX_1:def.12">MATRIX_1:def 12</a></span>
<br/>.= 
<font color="Maroon" title="c18">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E108">A49</font></span></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E91">A31</font></span></a>, <a class="txt" href="#E7:83_2"><span class="lab"><font color="Green" title="E92">A32</font></span></a>, <a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/><a NAME="E30:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E120">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><a NAME="E31:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E121">A67</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_2_2"><span class="lab"><font color="Green" title="E98">A39</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="E32:83_2_2_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">mSN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E15:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E109">A50</font></span></a>, <a class="txt" href="#E18:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E111">A52</font></span></a>, <a class="txt" href="#E30:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E120">A66</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><a NAME="E33:83_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A68</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">mSN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E34:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E123">A69</font></span>: 
 <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="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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="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="c4">A</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/><a NAME="E35:83_2_2_1_1_1"/>
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font></span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; 1 <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="#E4:83"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E18:83"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><a NAME="E36:83_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A70</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">x</font>,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font></span>)</span>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E18:83"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="txt" href="#E20:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E113">A54</font></span></a>, <a class="txt" href="#E12:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E107">A48</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A71</font></span>:  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font> = 
<span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<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>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E123">A69</font></span></a>, <a class="ref" href="matrix_3.html#T3" title="MATRIX_3:th.3">MATRIX_3:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c17">X</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E102">A43</font></span></a>, <a class="txt" href="#E6:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E103">A44</font></span></a>, <a class="txt" href="#E34:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E123">A69</font></span></a>, <a class="txt" href="#E36:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E124">A70</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c17">X</font>,1</span>)</span></span>)</span>

;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">mSN</font> = <font color="Maroon" title="c23">mSN</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E122">A68</font></span></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/><a NAME="E39:83_2_2_1_1_1"/><span class="lab"><font color="Green" title="E126">A72</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c12">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_2_1_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">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"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Maroon" title="c24">mSN</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:83_2_2_1_1_1_5"/><span class="lab"><font color="Green" title="E127">A73</font></span>: 
<font color="Maroon" title="c25">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"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Maroon" title="c24">mSN</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span><br/><span class="lab"><font color="Green" title="E128">A74</font></span>: <font color="Maroon" title="c25">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E127">A73</font></span></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></span>
;<br/><span class="lab"><font color="Green" title="E129">A75</font></span>:  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c25">j</font> = 
<font color="Maroon" title="c12">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E127">A73</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span> <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="c6">MV</font>,<font color="Maroon" title="c25">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_2"><span class="lab"><font color="Green" title="E94">A34</font></span></a>, <a class="txt" href="#E11:83_2"><span class="lab"><font color="Green" title="E95">A35</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E127">A73</font></span></a></span>
;<br/><a NAME="E4:83_2_2_1_1_1_5"/><span class="lab"><font color="Green" title="E130">A76</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E31:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E121">A67</font></span></a>, <a class="txt" href="#E30:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E120">A66</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E127">A73</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:83_2_2_1_1_1_5"/><span class="kw">then </span>
( <font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span> &amp; <font color="Maroon" title="c4">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span> <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="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E91">A31</font></span></a>, <a class="txt" href="#E7:83_2"><span class="lab"><font color="Green" title="E92">A32</font></span></a>, <a class="txt" href="#E2:83_2_2"><span class="lab"><font color="Green" title="E98">A39</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A77</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c20">x</font></span>)</span>,<font color="Maroon" title="c18">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E2:83_2_2"><span class="lab"><font color="Green" title="E98">A39</font></span></a>, <a class="txt" href="#E4:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E130">A76</font></span></a>, <a class="ref" href="fvsum_1.html#T74" title="FVSUM_1:th.74">FVSUM_1:74</a></span>
<br/>.= 
<font color="Maroon" title="c24">mSN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E18:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E111">A52</font></span></a>, <a class="txt" href="#E30:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E120">A66</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E127">A73</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/><span class="lab"><font color="Green" title="E132">A78</font></span>: <font color="Maroon" title="c20">x</font> = 
<span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E20:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E113">A54</font></span></a>, <a class="txt" href="#E12:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E107">A48</font></span></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></span>
;<br/><a NAME="E8:83_2_2_1_1_1_5"/><span class="lab"><font color="Green" title="E133">A79</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">N</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="c6">MV</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E14:83"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></span>;<br/><a NAME="E9:83_2_2_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">j</font>,<font color="Maroon" title="c16">i</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="c6">MV</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E14:83"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E100">A41</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E127">A73</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:83_2_2_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A80</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">j</font>,<font color="Maroon" title="c20">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">MV</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E106">A47</font></span></a>, <a class="txt" href="#E13:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E108">A49</font></span></a>, <a class="txt" href="#E9:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E105">A46</font></span></a>, <a class="txt" href="#E2:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E128">A74</font></span></a>, <a class="txt" href="#E8:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E133">A79</font></span></a>, <a class="ref" href="matrix13.html#T17" title="MATRIX13:th.17">MATRIX13:17</a></span>;<br/><a NAME="E11:83_2_2_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A81</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">j</font>,<font color="Maroon" title="c20">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="txt" href="matrix15.html#E3"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a></span>;<br/><a NAME="E12:83_2_2_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A82</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c25">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></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/><span class="lab"><font color="Green" title="E137">A83</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c25">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> = 
<font color="Maroon" title="c6">MV</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E13:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E108">A49</font></span></a>, <a class="txt" href="#E2:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E128">A74</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">j</font>,<font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="txt" href="#E10:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E134">A80</font></span></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">j</font>,<font color="Maroon" title="c20">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E135">A81</font></span></a>, <a class="ref" href="matrix_3.html#D2" title="MATRIX_3:def.2">MATRIX_3:def 2</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c25">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E136">A82</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E136">A82</font></span></a>, <a class="txt" href="#E7:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E132">A78</font></span></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</a></span>
;<br/><a NAME="E14:83_2_2_1_1_1_5"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_2"><span class="lab"><font color="Green" title="E93">A33</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> = 
<font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">j</font>,<font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E127">A73</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c25">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E39:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E126">A72</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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"><font color="Maroon" title="c4">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E100">A41</font></span></a>, <a class="txt" href="#E3:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E129">A75</font></span></a>, <a class="txt" href="#E13:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E137">A83</font></span></a>, <a class="ref" href="fvsum_1.html#T63" title="FVSUM_1:th.63">FVSUM_1:63</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T40" title="VECTSP_1:th.40">VECTSP_1:40</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Maroon" title="c24">mSN</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E127">A73</font></span></a>, <a class="txt" href="#E6:83_2_2_1_1_1_5"><span class="lab"><font color="Green" title="E131">A77</font></span></a>, <a class="ref" href="fvsum_1.html#T31" title="FVSUM_1:th.31">FVSUM_1:31</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E41:83_2_2_1_1_1"/><span class="kw">then </span>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Maroon" title="c24">mSN</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_2"><span class="lab"><font color="Green" title="E93">A33</font></span></a>, <a class="ref" href="finseq_2.html#T139" title="FINSEQ_2:th.139">FINSEQ_2:139</a></span>;<br/><span class="kw">hence </span> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c24">mSN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T94" title="FVSUM_1:th.94">FVSUM_1:94</a></span>
<br/>.= 
<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="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c24">mSN</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c24">mSN</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c22">mN</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E125">A71</font></span></a>, <a class="txt" href="#E7:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E104">A45</font></span></a>, <a class="txt" href="#E19:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E112">A53</font></span></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c24">mSN</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c24">mSN</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c22">mN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</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="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c22">mN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</a></span>
<br/>.= 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:83_2_2_1_1_1"><span class="lab"><font color="Green" title="E119">A65</font></span></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:83_2_2_1_1_2"/><span class="lab"><font color="Green" title="E100">A84</font></span>: 
not <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span><br/><div class="add"><a NAME="E2:83_2_2_1_1_2"/><span class="lab"><font color="Green" title="E101">A85</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_2_2"><span class="lab"><font color="Green" title="E98">A39</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="E3:83_2_2_1_1_2"/>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E100">A84</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:83_2_2_1_1_2"/><span class="lab"><font color="Green" title="E102">A86</font></span>: 
<font color="Maroon" title="c17">x</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:83_2_2_1_1_2"/><span class="lab"><font color="Green" title="E103">A87</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E101">A85</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c17">x</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="#E5:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E102">A86</font></span></a></span>;<br/><a NAME="E8:83_2_2_1_1_2"/><span class="lab"><font color="Green" title="E104">A88</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A89</font></span>:  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c18">x</font> = 
<font color="Maroon" title="c12">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E5:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E102">A86</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span></span>)</span> <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="c6">MV</font>,<font color="Maroon" title="c18">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E10:83_2"><span class="lab"><font color="Green" title="E94">A34</font></span></a>, <a class="txt" href="#E11:83_2"><span class="lab"><font color="Green" title="E95">A35</font></span></a>, <a class="txt" href="#E8:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E104">A88</font></span></a>, <a class="txt" href="#E5:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E102">A86</font></span></a></span>
;<br/><a NAME="E10:83_2_2_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E8:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E104">A88</font></span></a>, <a class="txt" href="#E5:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E102">A86</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E11:83_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A90</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c18">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/><font color="Maroon" title="c18">x</font> = 
<span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E8:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E104">A88</font></span></a>, <a class="txt" href="#E5:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E102">A86</font></span></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A91</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c18">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> = 
<font color="Maroon" title="c6">MV</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E6:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E103">A87</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E11:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E106">A90</font></span></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E106">A90</font></span></a>, <a class="ref" href="matrix_1.html#D12" title="MATRIX_1:def.12">MATRIX_1:def 12</a></span>
;<br/><a NAME="E14:83_2_2_1_1_2"/><span class="lab"><font color="Green" title="E108">A92</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#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/><a NAME="E15:83_2_2_1_1_2"/><span class="lab"><font color="Green" title="E109">A93</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M1</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="E16:83_2_2_1_1_2"/><span class="lab"><font color="Green" title="E110">A94</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><a NAME="E17:83_2_2_1_1_2"/><span class="lab"><font color="Green" title="E111">A95</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c12">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><div><span class="lab"><font color="Green" title="E112">A96</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_2_1_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">j</font> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83_2_2_1_1_2_4"/><span class="lab"><font color="Green" title="E113">A97</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:83_2_2_1_1_2_4"/><span class="lab"><font color="Green" title="E114">A98</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span><br/><span class="lab"><font color="Green" title="E115">A99</font></span>:  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c19">j</font> = 
<font color="Maroon" title="c12">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E108">A92</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_2_4"><span class="lab"><font color="Green" title="E113">A97</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span></span>)</span> <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="c6">MV</font>,<font color="Maroon" title="c19">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_2"><span class="lab"><font color="Green" title="E94">A34</font></span></a>, <a class="txt" href="#E14:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E108">A92</font></span></a>, <a class="txt" href="#E15:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E109">A93</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_2_4"><span class="lab"><font color="Green" title="E113">A97</font></span></a></span>
;<br/><a NAME="E4:83_2_2_1_1_2_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E8:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E104">A88</font></span></a>, <a class="txt" href="#E5:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E102">A86</font></span></a>, <a class="txt" href="#E14:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E108">A92</font></span></a>, <a class="txt" href="#E16:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E110">A94</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_2_4"><span class="lab"><font color="Green" title="E113">A97</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E5:83_2_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A100</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/><font color="Maroon" title="c19">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:83"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E14:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E108">A92</font></span></a>, <a class="txt" href="#E16:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E110">A94</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_2_4"><span class="lab"><font color="Green" title="E113">A97</font></span></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A101</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">MV</font>,<font color="Maroon" title="c19">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> = 
<font color="Maroon" title="c6">MV</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E6:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E103">A87</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E5:83_2_2_1_1_2_4"><span class="lab"><font color="Green" title="E116">A100</font></span></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</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="txt" href="#E2:83_2_2_1_1_2_4"><span class="lab"><font color="Green" title="E114">A98</font></span></a>, <a class="txt" href="#E5:83_2_2_1_1_2_4"><span class="lab"><font color="Green" title="E116">A100</font></span></a>, <a class="ref" href="matrix_1.html#D12" title="MATRIX_1:def.12">MATRIX_1:def 12</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> = 
<font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E108">A92</font></span></a>, <a class="txt" href="#E15:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E109">A93</font></span></a>, <a class="txt" href="#E1:83_2_2_1_1_2_4"><span class="lab"><font color="Green" title="E113">A97</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span></span>)</span> <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="c6">MV</font>,<font color="Maroon" title="c19">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E17:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E111">A95</font></span></a>, <a class="txt" href="#E3:83_2_2_1_1_2_4"><span class="lab"><font color="Green" title="E115">A99</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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="txt" href="#E12:83"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E7:83_2_2_1_1_2_4"><span class="lab"><font color="Green" title="E117">A101</font></span></a>, <a class="ref" href="fvsum_1.html#T63" title="FVSUM_1:th.63">FVSUM_1:63</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="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> = 
<font color="Maroon" title="c12">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E8:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E104">A88</font></span></a>, <a class="txt" href="#E5:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E102">A86</font></span></a>, <a class="txt" href="#E16:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E110">A94</font></span></a>, <a class="txt" href="#E15:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E109">A93</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c18">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E17:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E111">A95</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E13:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E107">A91</font></span></a>, <a class="txt" href="#E9:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E105">A89</font></span></a>, <a class="ref" href="fvsum_1.html#T63" title="FVSUM_1:th.63">FVSUM_1:63</a></span>
<br/>.= 
<font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E103">A87</font></span></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E91">A31</font></span></a>, <a class="txt" href="#E7:83_2"><span class="lab"><font color="Green" title="E92">A32</font></span></a>, <a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E20:83_2_2_1_1_2"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c12">M1</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:83"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E8:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E104">A88</font></span></a>, <a class="txt" href="#E5:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E102">A86</font></span></a>, <a class="txt" href="#E14:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E108">A92</font></span></a>, <a class="txt" href="#E16:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E110">A94</font></span></a>, <a class="txt" href="#E18:83_2_2_1_1_2"><span class="lab"><font color="Green" title="E112">A96</font></span></a>, <a class="ref" href="matrix_3.html#T14" title="MATRIX_3:th.14">MATRIX_3:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E4:83_2_2"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c13">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E5:83_2_2"/>
<font color="Maroon" title="c14">SumL</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:83"><span class="lab"><font color="Green" title="E87">A27</font></span></a>, <a class="txt" href="#E16:83_2"><span class="lab"><font color="Green" title="E96">A37</font></span></a>, <a class="txt" href="#E1:83_2_2"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E3:83_2_2"><span class="lab"><font color="Green" title="E99">A40</font></span></a>, <a class="ref" href="matrix13.html#T105" title="MATRIX13:th.105">MATRIX13:105</a>, <a class="ref" href="matrix13.html#T107" title="MATRIX13:th.107">MATRIX13:107</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E19:83_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A102</font></span>: 
<font color="Maroon" title="c14">SumL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T139" title="FINSEQ_2:th.139">FINSEQ_2:139</a></span>;<br/>
<a NAME="E20:83_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D2" title="VECTSP_7:def.2">VECTSP_7:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:83_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_2"><span class="lab"><font color="Green" title="E89">A29</font></span></a>, <a class="txt" href="#E19:83_2"><span class="lab"><font color="Green" title="E97">A102</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">take </span>
<font color="Maroon" title="c6">MV</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">MV</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">MV</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c4">A</font> )</span><br/>

<a NAME="E34:83"/>
 <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> <font color="Maroon" title="c4">A</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="c4">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP: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> 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="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="ref" href="matrix15.html#D5" target="_self" title="MATRIX15:def.5">Def5</a></span>;<br/>
<a NAME="E35:83"/><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="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span></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 <span class="p1">(<span class="default"><a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:83"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="ref" href="vectsp_7.html#T16" title="VECTSP_7:th.16">VECTSP_7:16</a></span>;<br/>
<a NAME="E36:83"/><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="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:83"><span class="lab"><font color="Green" title="E88">A28</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E37:83"/>
(  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">MV</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">MV</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c4">A</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">MV</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <font color="Maroon" title="c4">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:83"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E10:83"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="ref" href="vectsp_4.html#T37" title="VECTSP_4:th.37">VECTSP_4:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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