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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">n'</font>, <font color="Olive" title="b2">m'</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">n</font> being  <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#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Olive" title="b1">n'</font>,<font color="Olive" title="b2">m'</font>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b6">mt</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Olive" title="b3">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b7">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b8">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b9">nt</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Olive" title="b4">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st not <font color="Olive" title="b8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b9">nt</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b9">nt</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b6">mt</font></span>)</span></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="Olive" title="b5">A'</font> holds <br/> <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Olive" title="b5">A'</font>,<font color="Olive" title="b9">nt</font>,<font color="Olive" title="b6">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Olive" title="b5">A'</font>,<font color="Olive" title="b8">i</font>,<font color="Olive" title="b7">F</font></span>)</span>,<font color="Olive" title="b9">nt</font>,<font color="Olive" title="b6">mt</font></span><br/><b>let </b><font color="Maroon" title="c2">n'</font>, <font color="Maroon" title="c3">m'</font>, <font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">n</font> be   <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#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/> for <font color="Olive" title="b2">mt</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c4">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b5">nt</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st not <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b5">nt</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b5">nt</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">mt</font></span>)</span></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="Olive" title="b1">A'</font> holds <br/> <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b5">nt</font>,<font color="Olive" title="b2">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b3">F</font></span>)</span>,<font color="Olive" title="b5">nt</font>,<font color="Olive" title="b2">mt</font></span><br/><b>let </b><font color="Maroon" title="c6">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>; <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">mt</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c4">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b4">nt</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st not <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b4">nt</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b4">nt</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">mt</font></span>)</span></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="c6">A'</font> holds <br/> <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Olive" title="b4">nt</font>,<font color="Olive" title="b1">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">F</font></span>)</span>,<font color="Olive" title="b4">nt</font>,<font color="Olive" title="b1">mt</font></span><br/><b>let </b><font color="Maroon" title="c7">mt</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c4">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b3">nt</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st not <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">nt</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">nt</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">mt</font></span>)</span></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="c6">A'</font> holds <br/> <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Olive" title="b3">nt</font>,<font color="Maroon" title="c7">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b1">F</font></span>)</span>,<font color="Olive" title="b3">nt</font>,<font color="Maroon" title="c7">mt</font></span><br/><b>let </b><font color="Maroon" title="c8">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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"> for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">nt</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st not <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">nt</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">nt</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">mt</font></span>)</span></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="c6">A'</font> holds <br/> <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Olive" title="b2">nt</font>,<font color="Maroon" title="c7">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Olive" title="b2">nt</font>,<font color="Maroon" title="c7">mt</font></span><br/><b>let </b><font color="Maroon" title="c9">i</font> be   <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">nt</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st not <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">nt</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">nt</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">mt</font></span>)</span></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="c6">A'</font> holds <br/> <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Olive" title="b1">nt</font>,<font color="Maroon" title="c7">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Olive" title="b1">nt</font>,<font color="Maroon" title="c7">mt</font></span><br/><b>let </b><font color="Maroon" title="c10">nt</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">nt</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">nt</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">mt</font></span>)</span></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="c6">A'</font> implies  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font> )</span><br/>



















<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green" title="E38">A1</font></i>: 
not <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">nt</font>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green" title="E39">A2</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">nt</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">mt</font></span>)</span></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="c6">A'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span><br/>

<b>set </b><font color="Maroon" title="c11">R</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font>;<br/>
<b>set </b><font color="Maroon" title="c12">S</font> =  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font>;<br/>
<b>set </b><font color="Maroon" title="c13">SR</font> =  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">A'</font> or  <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="c6">A'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">A'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span><br/><div class="add"><b>hence </b><a NAME="E2:40_1_1"/>
 <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font>
 <b>by </b><i><a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/><i><font color="Green" title="E40">A3</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="c6">A'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span><br/><div class="add"><a NAME="E2:40_1_2"/><i><font color="Green" title="E41">A4</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">k</font>, <font color="Maroon" title="c15">m</font> be   <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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c15">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c15">m</font> )</span><br/><b>assume </b><a NAME="E1:40_1_2_1"/><i><font color="Green" title="E42">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c15">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c15">m</font></span><br/><b>reconsider </b><font color="Maroon" title="c16">K</font> = <font color="Maroon" title="c14">k</font>, <font color="Maroon" title="c17">M</font> = <font color="Maroon" title="c15">m</font> 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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:40_1_2_1"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">n</font></span>)</span>,<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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p5">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T26" title="MATRIX_1:th.26">MATRIX_1:26</a></i>;<br/><a NAME="E4:40_1_2_1"/><b>then </b>
( <font color="Maroon" title="c14">k</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> <font color="Maroon" title="c5">n</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="txt" href="matrix13.html#E1:40_1_2_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:40_1_2_1"/><b>then </b>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">nt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">nt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">mt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">M</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">A'</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:40"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="matrix13.html#E2:40"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="matrix13.html#E2:40_1_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="matrix13.html#E1:40_1_2_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="matrix13.html#T17" target="_self" title="MATRIX13:th.17">Th17</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:40_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">K</font>,<font color="Maroon" title="c17">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A'</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">nt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">mt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">M</font></span>)</span> &amp; <font color="Maroon" title="c6">A'</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">nt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">mt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">nt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">mt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">M</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:40_1_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="matrix13.html#E2:40_1_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="matrix13.html#E1:40_1_2_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="matrix13.html#D1" target="_self" title="MATRIX13:def.1">Def1</a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a></i>;<br/><b>hence </b><a NAME="E7:40_1_2_1"/>
<span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c15">m</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:40_1_2_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="matrix13.html#D1" target="_self" title="MATRIX13:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:40_1_2"/>
 <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c8">F</font></span>)</span>,<font color="Maroon" title="c10">nt</font>,<font color="Maroon" title="c7">mt</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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