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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">n'</font>, <font color="Maroon" title="c3">m'</font>, <font color="Maroon" title="c4">i</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c5">A'</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">n'</font>,<font color="Maroon" title="c3">m'</font>,<font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c6">Q</font>, <font color="Maroon" title="c7">P</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>let </b><font color="Maroon" title="c8">F</font>, <font color="Maroon" title="c9">FQ</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green" title="E60">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A'</font>
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c9">FQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q</font></span>)</span>
 <b>and </b><br/><a NAME="E3:68"/><i><font color="Green" title="E62">A3</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">P</font>,<font color="Maroon" title="c6">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="c5">A'</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c10">SP</font> =  <a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font>;<br/>
<b>set </b><font color="Maroon" title="c11">SQ</font> =  <a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q</font>;<br/>
<b>consider </b><font color="Maroon" title="c12">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="E5:68"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c7">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="c12">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="c13">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="E7:68"/><i><font color="Green" title="E64">A5</font></i>: 
<font color="Maroon" title="c6">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="c13">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="E8:68"/><i><font color="Green" title="E65">A6</font></i>: 
(  <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="c7">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c6">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q</font> &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">P</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c7">P</font>,<font color="Maroon" title="c6">Q</font></span>)</span> &amp;  <a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">without_repeated_line</a> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E5:68"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="matrix13.html#E7:68"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a>, <a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font></span>)</span> or  not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1"><b>suppose </b></a><a NAME="E1:68_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:68_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is    <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>by </b><i><a class="txt" href="matrix13.html#E8:68"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="finseq_5.html#T4" title="FINSEQ_5:th.4">FINSEQ_5:4</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E3:68_1_1"/>
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c7">P</font>,<font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">FQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c5">A'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c7">P</font>,<font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T37" target="_self" title="MATRIX13:th.37">TT37</a>, <a class="txt" href="matrix13.html#E1:68"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="matrix13.html#E2:68"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="matrix13.html#E3:68"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="matrix13.html#E8:68"><i><font color="Green" title="E65">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2"><b>suppose </b></a><a NAME="E1:68_1_2"/><i><font color="Green" title="E66">A7</font></i>: 
not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:68_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A'</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b> <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c5">A'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c7">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c7">P</font>,<font color="Maroon" title="c6">Q</font> = 
 <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c7">P</font>,<font color="Maroon" title="c6">Q</font>
<b>by </b><i><a class="ref" href="matrix13.html#T40" target="_self" title="MATRIX13:th.40">TT40</a></i>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c7">P</font>,<font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">FQ</font>
<b>by </b><i><a class="txt" href="matrix13.html#E8:68"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="matrix13.html#E1:68_1_2"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="matrix13.html#T40" target="_self" title="MATRIX13:th.40">TT40</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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