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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</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> 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><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>  <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="Olive" title="b2">P</font></span>)</span> &amp; <font color="Olive" title="b5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">P</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">Q</font> holds <br/>(  <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="Olive" title="b1">M</font>,<font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">Q</font>)</span>)</span>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Olive" title="b2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Olive" title="b3">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>) &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Olive" title="b2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font></span>)</span></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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Olive" title="b3">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a><br/>  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">P</font></span>)</span> &amp; <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">P</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">Q</font> holds <br/>(  <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">M</font>,<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Olive" title="b1">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Olive" title="b2">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>) &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Olive" title="b1">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span></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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Olive" title="b2">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">P1</font>, <font color="Maroon" title="c4">Q1</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Q1</font> holds <br/>(  <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">M</font>,<font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font>)</span>)</span>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>) &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span> &amp; <font color="Maroon" title="c6">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Q1</font> implies (  <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">M</font>,<font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font>)</span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>) &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:75"/><span class="lab"><font color="Green" title="E65">A1</font></span>: 
<font color="Maroon" title="c5">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:75"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
<font color="Maroon" title="c6">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:75"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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">M</font>,<font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font>)</span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>) &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">SQ1</font> =  <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font>;<br/>
<a NAME="E4:75"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c4">Q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></span>;<br/>
<a NAME="E5:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A5</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="c4">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Q1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T40" title="FINSEQ_3:th.40">FINSEQ_3:40</a></span>;<br/>
<a NAME="E6:75"/><span class="lab"><font color="Green" title="E70">A6</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="c4">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/>
<a NAME="E7:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A7</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">Q1</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="relat_1.html#T134" title="RELAT_1:th.134">RELAT_1:134</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Q</font> = <font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">Q2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Q1</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c6">j</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>;<br/>
<a NAME="E8:75"/><span class="lab"><font color="Green" title="E72">A8</font></span>: 
<font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">SP1</font> =  <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font>;<br/>
<a NAME="E9:75"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c3">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></span>;<br/>
<a NAME="E10:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A10</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="c3">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T40" title="FINSEQ_3:th.40">FINSEQ_3:40</a></span>;<br/>
<a NAME="E11:75"/><span class="lab"><font color="Green" title="E75">A11</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="c3">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:75"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/>
<a NAME="E12:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A12</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c3">P1</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:75"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="ref" href="relat_1.html#T134" title="RELAT_1:th.134">RELAT_1:134</a></span>;<br/>
<a NAME="E13:75"/>
 <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font> is  <a href="matrix13.html#NV1" title="MATRIX13:NV.1">without_repeated_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="finseq_3.html#T92" title="FINSEQ_3:th.92">FINSEQ_3:92</a></span>;<br/>
<a NAME="E14:75"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c6">j</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="finseq_5.html#T4" title="FINSEQ_5:th.4">FINSEQ_5:4</a></span>;<br/>
<a NAME="E15:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A13</font></span>: 
<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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Q1</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c6">j</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <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">Q1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></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="#E3:75"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="funct_1.html#T69" title="FUNCT_1:th.69">FUNCT_1:69</a></span>;<br/>
<a NAME="E16:75"/><span class="lab"><font color="Green" title="E78">A14</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E10:75"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E11:75"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">P2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c5">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">P</font> = <font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>;<br/>
<a NAME="E17:75"/>
 <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font> is  <a href="matrix13.html#NV1" title="MATRIX13:NV.1">without_repeated_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:75"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="ref" href="finseq_3.html#T92" title="FINSEQ_3:th.92">FINSEQ_3:92</a></span>;<br/>
<a NAME="E18:75"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c5">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E10:75"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="ref" href="finseq_5.html#T4" title="FINSEQ_5:th.4">FINSEQ_5:4</a></span>;<br/>
<a NAME="E19:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A15</font></span>: 
<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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c5">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <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">P1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></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="#E12:75"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="ref" href="funct_1.html#T69" title="FUNCT_1:th.69">FUNCT_1:69</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">E</font> =  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font>);<br/>
<a NAME="E20:75"/><span class="lab"><font color="Green" title="E80">A16</font></span>: 
<font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E21:75"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E65">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">C</font> = <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/>
<a NAME="E23:75"/><span class="lab"><font color="Green" title="E81">A17</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">C</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E24:75"/>
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E6:75"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E25:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A18</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="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E23:75"><span class="lab"><font color="Green" title="E81">A17</font></span></a>, <a class="ref" href="stirl2_1.html#T55" title="STIRL2_1:th.55">STIRL2_1:55</a></span>;<br/>
 <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">M</font>,<font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font>)</span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>) = 
 <a href="matrix_2.html#K9" title="MATRIX_2:func.9">Deleting</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font>)</span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="laplace.html#D1" title="LAPLACE:def.1">LAPLACE:def 1</a></span>
<br/>.= 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c5">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"><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> <font color="Maroon" title="c4">Q1</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c6">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="#E3:75"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="matrix13.html#T58" target="_self" title="MATRIX13:th.58">Th58</a></span>
<br/>.= 
 <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">M</font>,<font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P1</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c5">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"><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> <font color="Maroon" title="c4">Q1</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c6">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="#E3:75"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
<br/>.= 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></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="#E20:75"><span class="lab"><font color="Green" title="E80">A16</font></span></a>, <a class="txt" href="#E8:75"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="txt" href="#E19:75"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="txt" href="#E15:75"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="ref" href="matrix13.html#T56" target="_self" title="MATRIX13:th.56">Th56</a></span>
<br/>.= 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></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="#E16:75"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E23:75"><span class="lab"><font color="Green" title="E81">A17</font></span></a>, <a class="txt" href="#E25:75"><span class="lab"><font color="Green" title="E82">A18</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a>, <a class="ref" href="stirl2_1.html#T55" title="STIRL2_1:th.55">STIRL2_1:55</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E27:75"/>
(  <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">M</font>,<font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font>)</span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>) &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c3">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></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="#E16:75"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E23:75"><span class="lab"><font color="Green" title="E81">A17</font></span></a>, <a class="txt" href="#E25:75"><span class="lab"><font color="Green" title="E82">A18</font></span></a>, <a class="ref" href="stirl2_1.html#T55" title="STIRL2_1:th.55">STIRL2_1:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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