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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> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</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> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:71"/><i><font color="Green" title="E62">A1</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green" title="E63">A2</font></i>: 
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:71"/><i><font color="Green" title="E64">A3</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">TT43</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:71"/><i><font color="Green" title="E65">A4</font></i>: 
<font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">TT43</a></i>;<br/>
<a NAME="E7:71"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E4:71"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="matrix13.html#E6:71"><i><font color="Green" title="E65">A4</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E8:71"/>
 <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
 <b>by </b><i><a class="txt" href="matrix13.html#E1:71"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="matrix13.html#E2:71"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="matrix13.html#T19" target="_self" title="MATRIX13:th.19">TT19</a></i>;<br/>


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