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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">Q</font>, <font color="Olive" title="b4">Q9</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 <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">P</font>,<font color="Olive" title="b4">Q9</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="Olive" title="b1">A</font> holds <br/>for <font color="Olive" title="b5">i</font>, <font color="Olive" title="b6">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b6">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b3">Q</font> &amp; <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b6">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Olive" title="b3">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">Q9</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b5">i</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Olive" title="b4">Q9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b4">Q9</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">Q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font>, <font color="Olive" title="b3">Q9</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 <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">P</font>,<font color="Olive" title="b3">Q9</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="c2">A</font> holds <br/>for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b5">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Olive" title="b2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Q9</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b4">i</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Olive" title="b3">Q9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">Q9</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font>, <font color="Maroon" title="c5">R</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"> ( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">R</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="c2">A</font> implies for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b2">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">i</font></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">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">R</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">R</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="c2">A</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">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b2">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">i</font></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">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">R</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c7">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">R</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> implies  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:12"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E3:12"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c7">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">and </span><br/><a NAME="E4:12"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">and </span><br/><a NAME="E5:12"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font>
 <span class="kw">and </span><br/><a NAME="E6:12"/><span class="lab"><font color="Green" title="E18">A6</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="c2">A</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">R</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span><br/>

<a NAME="E7:12"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<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="E8:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E9:12"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">R</font></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="#E5:12"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></span>;<br/>
<a NAME="E10:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">i0</font> = <font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c9">j0</font> = <font color="Maroon" title="c7">j</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E19">A7</font></span></a></span>;<br/>
<a NAME="E12:12"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E15">A3</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">set </span><font color="Maroon" title="c10">S</font> =  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">P9</font>, <font color="Maroon" title="c12">Q9</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E14:12"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c11">P9</font>,<font color="Maroon" title="c12">Q9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:12"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">Q9</font>
 <span class="kw">and </span><br/><a NAME="E16:12"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:12"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span>,<font color="Maroon" title="c11">P9</font>,<font color="Maroon" title="c12">Q9</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#D4" title="MATRIX13:def.4">MATRIX13:def 4</a></span>;<br/>
<a NAME="E18:12"/>
( <font color="Maroon" title="c11">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Maroon" title="c12">Q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E15:12"><span class="lab"><font color="Green" title="E25">A13</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">P2</font>, <font color="Maroon" title="c14">Q2</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E20:12"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<font color="Maroon" title="c13">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E21:12"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
<font color="Maroon" title="c14">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">and </span><br/><a NAME="E22:12"/>
<font color="Maroon" title="c13">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">P9</font>
 <span class="kw">and </span><br/><a NAME="E23:12"/>
<font color="Maroon" title="c14">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c12">Q9</font>
 <span class="kw">and </span><br/><a NAME="E24:12"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">P9</font>
 <span class="kw">and </span><br/><a NAME="E25:12"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">Q9</font>
 <span class="kw">and </span><br/><a NAME="E26:12"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span>,<font color="Maroon" title="c11">P9</font>,<font color="Maroon" title="c12">Q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c13">P2</font>,<font color="Maroon" title="c14">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:12"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="matrix13.html#T57" title="MATRIX13:th.57">MATRIX13:57</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">Q2j</font> = <font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">P2i</font> = <font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">ESS</font> =  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">SS</font> =  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_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="E1:12_1"/>
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><span class="kw">suppose </span></a><a NAME="E1:12_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span><br/><div class="add"><a NAME="E2:12_1_1"/><span class="kw">then </span>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></span>;<br/><a NAME="E3:12_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A21</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T77" title="MATRIX13:th.77">MATRIX13:77</a></span>;<br/><a NAME="E4:12_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E23">A11</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">hence </span><a NAME="E5:12_1_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E3:12_1_1"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="matrix13.html#T94" title="MATRIX13:th.94">MATRIX13:94</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2"><span class="kw">suppose </span></a><a NAME="E1:12_1_2"/><span class="lab"><font color="Green" title="E33">A22</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span><br/><div class="add"><a NAME="E2:12_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T138" title="ZFMISC_1:th.138">ZFMISC_1:138</a></span>;<br/><a NAME="E3:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A23</font></span>: 
<font color="Maroon" title="c13">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:12"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E4:12_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <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="#E9:12"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E1:12_1_2"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></span>;<br/><a NAME="E5:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A24</font></span>: 
<font color="Maroon" title="c5">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T138" title="ZFMISC_1:th.138">ZFMISC_1:138</a></span>;<br/><a NAME="E6:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A25</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">R</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">R</font></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></span>;<br/><a NAME="E7:12_1_2"/>
<font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E1:12_1_2"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T138" title="ZFMISC_1:th.138">ZFMISC_1:138</a></span>;<br/><a NAME="E8:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A26</font></span>: 
<font color="Maroon" title="c14">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:12"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E9:12_1_2"/><span class="lab"><font color="Green" title="E38">A27</font></span>: 
<span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E10:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A28</font></span>: 
 <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_1_2"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E11:12_1_2"/><span class="lab"><font color="Green" title="E40">A29</font></span>: 
<font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_1_2"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="txt" href="#E9:12_1_2"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E12:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A30</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"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><a NAME="E13:12_1_2"/><span class="lab"><font color="Green" title="E42">A31</font></span>: 
<span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E14:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A32</font></span>: 
<font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_2"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E15:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A33</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"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_1_2"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></span>;<br/><a NAME="E16:12_1_2"/><span class="lab"><font color="Green" title="E45">A34</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"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E3:12_1_2"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="txt" href="#E13:12_1_2"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E17:12_1_2"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c6">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E18:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A35</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E19:12_1_2"/><span class="lab"><font color="Green" title="E47">A36</font></span>: 
not <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E20:12"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E20:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A37</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">P2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></span>;<br/><a NAME="E21:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A38</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E22:12_1_2"/><span class="lab"><font color="Green" title="E50">A39</font></span>: 
not <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E21:12"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E23:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A40</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">Q2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></span>;<br/><a NAME="E24:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A41</font></span>: 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E25:12"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E19:12_1_2"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a>, <a class="ref" href="matrix13.html#D3" title="MATRIX13:def.3">MATRIX13:def 3</a></span>;<br/><a NAME="E25:12_1_2"/>
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c7">j</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E26:12_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E28:12_1_2"/><span class="lab"><font color="Green" title="E53">A42</font></span>: 
<font color="Maroon" title="c19">y</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"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E29:12_1_2"/><span class="lab"><font color="Green" title="E54">A43</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"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:12_1_2"><span class="lab"><font color="Green" title="E41">A30</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="E30:12_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E14:12_1_2"><span class="lab"><font color="Green" title="E43">A32</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="c20">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E32:12_1_2"/><span class="lab"><font color="Green" title="E55">A44</font></span>: 
<font color="Maroon" title="c20">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"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E33:12_1_2"/><span class="lab"><font color="Green" title="E56">A45</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"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:12_1_2"><span class="lab"><font color="Green" title="E46">A35</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="c21">x</font> = <font color="Maroon" title="c20">x</font>, <font color="Maroon" title="c22">y</font> = <font color="Maroon" title="c19">y</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="#E32:12_1_2"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E28:12_1_2"><span class="lab"><font color="Green" title="E53">A42</font></span></a></span>;<br/><a NAME="E35:12_1_2"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T86" title="VECTSP_1:th.86">VECTSP_1:86</a></span>;<br/><a NAME="E36:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A46</font></span>: 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">y</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="matrix15.html#E4"><span class="lab"><font color="Green" title="E4">Lm2</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c23">L</font> =  <a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font>;<br/><span class="lab"><font color="Green" title="E58">A47</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p5"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p5"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="laplace.html#D7" title="LAPLACE:def.7">LAPLACE:def 7</a></span>
;<br/><a NAME="E38:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A48</font></span>: 
<font color="Maroon" title="c22">y</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="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E25:12"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E8:12_1_2"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="txt" href="#E9:12_1_2"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="txt" href="#E20:12_1_2"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="txt" href="#E23:12_1_2"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="txt" href="#E28:12_1_2"><span class="lab"><font color="Green" title="E53">A42</font></span></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E39:12_1_2"/><span class="lab"><font color="Green" title="E60">A49</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"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_1_2"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="txt" href="#E9:12_1_2"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><span class="kw">then </span> <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c22">y</font> = 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c6">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c7">j</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E25:12"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E20:12_1_2"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="txt" href="#E23:12_1_2"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="txt" href="#E16:12_1_2"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="txt" href="#E32:12_1_2"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E33:12_1_2"><span class="lab"><font color="Green" title="E56">A45</font></span></a>, <a class="txt" href="#E28:12_1_2"><span class="lab"><font color="Green" title="E53">A42</font></span></a>, <a class="txt" href="#E29:12_1_2"><span class="lab"><font color="Green" title="E54">A43</font></span></a>, <a class="ref" href="matrix13.html#T64" title="MATRIX13:th.64">MATRIX13:64</a></span>
<br/>.= 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c13">P2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c7">j</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:12_1_2"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="ref" href="zfmisc_1.html#T141" title="ZFMISC_1:th.141">ZFMISC_1:141</a></span>
<br/>.= 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c13">P2</font>,<font color="Maroon" title="c14">Q2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:12_1_2"><span class="lab"><font color="Green" title="E50">A39</font></span></a>, <a class="ref" href="zfmisc_1.html#T141" title="ZFMISC_1:th.141">ZFMISC_1:141</a></span>
<br/>.= 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c13">P2</font>,<font color="Maroon" title="c14">Q2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E25:12"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="matrix13.html#D3" title="MATRIX13:def.3">MATRIX13:def 3</a></span>
<br/>.= 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span>,<font color="Maroon" title="c11">P9</font>,<font color="Maroon" title="c12">Q9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E26:12"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="matrix13.html#D3" title="MATRIX13:def.3">MATRIX13:def 3</a></span>
;<br/><a NAME="E41:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A50</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <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="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">y</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="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p5"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p5"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c22">y</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:12"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E21:12_1_2"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="txt" href="#E36:12_1_2"><span class="lab"><font color="Green" title="E57">A46</font></span></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></span>;<br/><a NAME="E42:12_1_2"/><span class="lab"><font color="Green" title="E62">A51</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p5"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p5"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/><a NAME="E43:12_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A52</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c22">y</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="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E25:12"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E20:12_1_2"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="txt" href="#E23:12_1_2"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="txt" href="#E16:12_1_2"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="txt" href="#E39:12_1_2"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="txt" href="#E32:12_1_2"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E28:12_1_2"><span class="lab"><font color="Green" title="E53">A42</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E44:12_1_2"/><span class="lab"><font color="Green" title="E64">A53</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">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_1_2"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">k</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">k</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="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span> &amp; <font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">y</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:12_1_2_3"/><span class="lab"><font color="Green" title="E65">A54</font></span>: 
<font color="Maroon" title="c24">k</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="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:12_1_2_3"/><span class="lab"><font color="Green" title="E66">A55</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">y</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="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span><br/><a NAME="E3:12_1_2_3"/>
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E25:12"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E20:12_1_2"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="txt" href="#E23:12_1_2"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="txt" href="#E39:12_1_2"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="txt" href="#E10:12_1_2"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="txt" href="#E28:12_1_2"><span class="lab"><font color="Green" title="E53">A42</font></span></a>, <a class="txt" href="#E29:12_1_2"><span class="lab"><font color="Green" title="E54">A43</font></span></a>, <a class="txt" href="#E37:12_1_2"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="txt" href="#E1:12_1_2_3"><span class="lab"><font color="Green" title="E65">A54</font></span></a>, <a class="txt" href="#E2:12_1_2_3"><span class="lab"><font color="Green" title="E66">A55</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><a NAME="E4:12_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A56</font></span>: 
not <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c7">j</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:12_1_2_3"/>
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E25:12"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E20:12_1_2"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="txt" href="#E23:12_1_2"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="txt" href="#E12:12_1_2"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E39:12_1_2"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="txt" href="#E37:12_1_2"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="txt" href="#E1:12_1_2_3"><span class="lab"><font color="Green" title="E65">A54</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="E6:12_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1_2_3"><span class="lab"><font color="Green" title="E67">A56</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:12_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A57</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"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:12"><span class="lab"><font color="Green" title="E29">A17</font></span></a></span>;<br/><a NAME="E8:12_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A58</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"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12"><span class="lab"><font color="Green" title="E17">A5</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c25">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:12_1_2_3"/><span class="lab"><font color="Green" title="E70">A59</font></span>: 
<font color="Maroon" title="c25">z</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">R</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:12_1_2_3"/><span class="lab"><font color="Green" title="E71">A60</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.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"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E44:12_1_2"><span class="lab"><font color="Green" title="E64">A53</font></span></a>, <a class="txt" href="#E7:12_1_2_3"><span class="lab"><font color="Green" title="E68">A57</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="c26">z</font> = <font color="Maroon" title="c25">z</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="#E10:12_1_2_3"><span class="lab"><font color="Green" title="E70">A59</font></span></a></span>;<br/><a NAME="E13:12_1_2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c24">k</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#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:12_1_2"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="txt" href="#E32:12_1_2"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E37:12_1_2"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="txt" href="#E42:12_1_2"><span class="lab"><font color="Green" title="E62">A51</font></span></a>, <a class="txt" href="#E1:12_1_2_3"><span class="lab"><font color="Green" title="E65">A54</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="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c24">k</font> = 
<font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</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"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:12_1_2"><span class="lab"><font color="Green" title="E56">A45</font></span></a>, <a class="txt" href="#E24:12_1_2"><span class="lab"><font color="Green" title="E52">A41</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"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">i</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">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_1_2"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="txt" href="#E11:12_1_2_3"><span class="lab"><font color="Green" title="E71">A60</font></span></a>, <a class="txt" href="#E8:12_1_2_3"><span class="lab"><font color="Green" title="E69">A58</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"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">R</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="c1">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E10:12_1_2_3"><span class="lab"><font color="Green" title="E70">A59</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_1_2"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="txt" href="#E10:12_1_2_3"><span class="lab"><font color="Green" title="E70">A59</font></span></a>, <a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c24">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_2_3"><span class="lab"><font color="Green" title="E65">A54</font></span></a>, <a class="ref" href="laplace.html#D7" title="LAPLACE:def.7">LAPLACE:def 7</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</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><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A61</font></span>: <span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:12_1_2"><span class="lab"><font color="Green" title="E59">A48</font></span></a>, <a class="ref" href="matrix_3.html#T14" title="MATRIX_3:th.14">MATRIX_3:14</a></span>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:12_1_2"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="txt" href="#E32:12_1_2"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="ref" href="laplace.html#T25" title="LAPLACE:th.25">LAPLACE:25</a></span>
;<br/><span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font> = 
<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="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c22">y</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c22">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:12_1_2"><span class="lab"><font color="Green" title="E59">A48</font></span></a>, <a class="txt" href="#E24:12_1_2"><span class="lab"><font color="Green" title="E52">A41</font></span></a>, <a class="ref" href="laplace.html#D7" title="LAPLACE:def.7">LAPLACE:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></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"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">y</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p0"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p0"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c22">y</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:12_1_2"><span class="lab"><font color="Green" title="E56">A45</font></span></a>, <a class="txt" href="#E29:12_1_2"><span class="lab"><font color="Green" title="E54">A43</font></span></a>, <a class="txt" href="#E24:12_1_2"><span class="lab"><font color="Green" title="E52">A41</font></span></a>, <a class="txt" href="#E43:12_1_2"><span class="lab"><font color="Green" title="E63">A52</font></span></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</a></span>
;<br/><a NAME="E48:12_1_2"/><span class="kw">then </span>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c9">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E46:12_1_2"><span class="lab"><font color="Green" title="E65">A61</font></span></a>, <a class="txt" href="#E41:12_1_2"><span class="lab"><font color="Green" title="E61">A50</font></span></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></span>;<br/><a NAME="E49:12_1_2"/><span class="kw">then </span>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c8">i0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E25:12"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E20:12_1_2"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="txt" href="#E23:12_1_2"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="txt" href="#E15:12_1_2"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="ref" href="matrix13.html#D4" title="MATRIX13:def.4">MATRIX13:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E50:12_1_2"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:12"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E20:12_1_2"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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>
