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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>











<b>assume </b><a NAME="E1:54"/>
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:54"/><b>then </b><i><font color="Green" title="E45">A1</font></i>: 
(  <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 &amp;  <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 &amp;  <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> &amp;  <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> )
 
<b>by </b><i><a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a>, <a class="ref" href="finseq_3.html#T51" title="FINSEQ_3:th.51">FINSEQ_3:51</a></i>;<br/>
<a NAME="E3:54"/><b>then </b>
( <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> &amp; <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> &amp; <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> &amp; <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> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E4:54"/>
 <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>
 <b>by </b><i><a class="txt" href="#E2:54"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="matrix13.html#T23" target="_self" title="MATRIX13:th.23">Th23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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