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<span class="kw">let </span><font color="Maroon" title="c1">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">i0</font>, <font color="Olive" title="b2">j0</font>, <font color="Olive" title="b3">n0</font>, <font color="Olive" title="b4">m0</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b5">A</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j0</font> &amp; <font color="Olive" title="b3">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">m0</font> holds <br/> <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b5">A</font>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Olive" title="b1">i0</font>,<font color="Olive" title="b2">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Olive" title="b3">n0</font>,<font color="Olive" title="b4">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i0</font>,<font color="Olive" title="b3">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b5">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i0</font>,<font color="Olive" title="b4">m0</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j0</font>,<font color="Olive" title="b3">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b5">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j0</font>,<font color="Olive" title="b4">m0</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">i0</font>, <font color="Maroon" title="c3">j0</font>, <font color="Maroon" title="c4">n0</font>, <font color="Maroon" title="c5">m0</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c2">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j0</font> &amp; <font color="Maroon" title="c4">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m0</font> holds <br/> <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b1">A</font>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c3">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c4">n0</font>,<font color="Maroon" title="c5">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c4">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c5">m0</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">j0</font>,<font color="Maroon" title="c4">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">j0</font>,<font color="Maroon" title="c5">m0</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j0</font> &amp; <font color="Maroon" title="c4">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m0</font> implies  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">A</font>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c3">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c4">n0</font>,<font color="Maroon" title="c5">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c4">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c5">m0</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">j0</font>,<font color="Maroon" title="c4">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">j0</font>,<font color="Maroon" title="c5">m0</font></span>)</span> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c2">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j0</font>
 <span class="kw">and </span><br/><a NAME="E2:54"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c4">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">A</font>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c3">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c4">n0</font>,<font color="Maroon" title="c5">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c4">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c5">m0</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">j0</font>,<font color="Maroon" title="c4">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">j0</font>,<font color="Maroon" title="c5">m0</font></span>)</span></span><br/>

<a NAME="E3:54"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c4">n0</font>,<font color="Maroon" title="c5">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></span>;<br/>
<a NAME="E4:54"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
 <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c4">n0</font>,<font color="Maroon" title="c5">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">n0</font>,<font color="Maroon" title="c5">m0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="finseq_3.html#T51" title="FINSEQ_3:th.51">FINSEQ_3:51</a></span>;<br/>
<a NAME="E5:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A5</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c4">n0</font>,<font color="Maroon" title="c5">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/>
<a NAME="E6:54"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
 <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c3">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c3">j0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="finseq_3.html#T51" title="FINSEQ_3:th.51">FINSEQ_3:51</a></span>;<br/>
<a NAME="E7:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c3">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/>
<a NAME="E8:54"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c3">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/>
<a NAME="E9:54"/><span class="lab"><font color="Green" title="E53">A9</font></span>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c4">n0</font>,<font color="Maroon" title="c5">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/>
<a NAME="E10:54"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c3">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:54"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">A</font>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c3">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c4">n0</font>,<font color="Maroon" title="c5">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c4">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i0</font>,<font color="Maroon" title="c5">m0</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">j0</font>,<font color="Maroon" title="c4">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">j0</font>,<font color="Maroon" title="c5">m0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="ref" href="matrix13.html#T23" target="_self" title="MATRIX13:th.23">Th23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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