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<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> ; <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">lk</font> being   <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> <br/> for <font color="Olive" title="b2">Key</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Olive" title="b1">lk</font>,6, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">r</font> being   <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> <br/> for <font color="Olive" title="b4">M</font>, <font color="Olive" title="b5">m</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Olive" title="b2">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b5">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b5">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b5">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b5">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> holds <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b4">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><b>let </b><font color="Maroon" title="c2">lk</font> be    <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> ; <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">Key</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">lk</font>,6, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">r</font> being   <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> <br/> for <font color="Olive" title="b3">M</font>, <font color="Olive" title="b4">m</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Olive" title="b1">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b4">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> holds <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b3">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><b>let </b><font color="Maroon" title="c3">Key</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">lk</font>,6, <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">r</font> being   <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> <br/> for <font color="Olive" title="b2">M</font>, <font color="Olive" title="b3">m</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b3">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> holds <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><b>let </b><font color="Maroon" title="c4">r</font> be    <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> ; <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">M</font>, <font color="Olive" title="b2">m</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> holds <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b1">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><b>let </b><font color="Maroon" title="c5">M</font>, <font color="Maroon" title="c6">m</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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"> ( <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c6">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> implies (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> ) )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:76"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
 <b>and </b><br/><a NAME="E2:76"/><i><font color="Green" title="E60">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c6">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/>

<a NAME="E3:76"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1"/>
( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1"><b>suppose </b></a><a NAME="E1:76_1_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><div class="add"><a NAME="E2:76_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></i>;<br/><a NAME="E3:76_1_1"/><b>then </b>
 <a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><b>then </b><font color="Maroon" title="c5">M</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:76"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="funct_7.html#T41" title="FUNCT_7:th.41">FUNCT_7:41</a></i>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="txt" href="idea_1.html#E3:76"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/><b>hence </b><a NAME="E5:76_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="idea_1.html#E2:76"><i><font color="Green" title="E60">A2</font></i></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="76_1_2"><b>suppose </b></a><a NAME="E1:76_1_2"/><i><font color="Green" title="E62">A4</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c7">r1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:76_1_2"/><i><font color="Green" title="E63">A5</font></i>: 
<font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E4:76_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:76_1_2"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E5:76_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">r1</font> = <font color="Maroon" title="c7">r1</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="txt" href="idea_1.html#E3:76_1_2"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E7:76_1_2"/>
 <a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">r1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="idea_1.html#T37" target="_self" title="IDEA_1:th.37">Th37</a></i>;<br/><a NAME="E8:76_1_2"/><b>then </b><i><font color="Green" title="E64">A6</font></i>: 
 <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">r1</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:76_1_2"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="funct_7.html#T43" title="FUNCT_7:th.43">FUNCT_7:43</a></i>;<br/><a NAME="E9:76_1_2"/><b>then </b>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">r1</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:76"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="idea_1.html#T44" target="_self" title="IDEA_1:th.44">Th44</a></i>;<br/><a NAME="E10:76_1_2"/><b>then </b><i><font color="Green" title="E65">A7</font></i>: 
<font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">r1</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:76"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="idea_1.html#E8:76_1_2"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E11:76_1_2"/><i><font color="Green" title="E66">A8</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">r1</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">r1</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  )
 <b>by </b><i><a class="ref" href="idea_1.html#T44" target="_self" title="IDEA_1:th.44">Th44</a></i>;<br/><a NAME="E12:76_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">r1</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</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> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">r1</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:76"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">M1</font> = <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">r1</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="idea_1.html#E11:76_1_2"><i><font color="Green" title="E66">A8</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E14:76_1_2"/><i><font color="Green" title="E67">A9</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>for <font color="Olive" title="b1">M</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,$1</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4;<br/><a NAME="E15:76_1_2"/><i><font color="Green" title="E68">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">M</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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"> ( <font color="Maroon" title="c10">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 )</span><br/>

<b>assume </b><a NAME="E1:76_1_2_1"/><i><font color="Green" title="E69">A11</font></i>: 
<font color="Maroon" title="c10">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4</span><br/>

<a NAME="E2:76_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></i>;<br/>
<a NAME="E3:76_1_2_1"/><b>then </b>
 <a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<b>then </b><font color="Maroon" title="c10">M</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:76_1_2_1"><i><font color="Green" title="E69">A11</font></i></a>, <a class="ref" href="funct_7.html#T41" title="FUNCT_7:th.41">FUNCT_7:41</a></i>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="txt" href="idea_1.html#E14:76_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>
<b>hence </b><a NAME="E5:76_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 <b>by </b><i><a class="txt" href="idea_1.html#E2:76"><i><font color="Green" title="E60">A2</font></i></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><a NAME="E16:76_1_2"/><i><font color="Green" title="E69">A12</font></i>: 
for <font color="Olive" title="b1">k</font> being   <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:76_1_2_2"/><i><font color="Green" title="E70">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>let </b><font color="Maroon" title="c11">M</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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"> ( <font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 )</span><br/>

<b>assume </b><a NAME="E2:76_1_2_2"/>
<font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4</span><br/>

<a NAME="E3:76_1_2_2"/><b>then </b>
<font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="ref" href="idea_1.html#T37" target="_self" title="IDEA_1:th.37">Th37</a></i>;<br/>
<a NAME="E4:76_1_2_2"/><b>then </b><i><font color="Green" title="E71">A14</font></i>: 
<font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T43" title="FUNCT_7:th.43">FUNCT_7:43</a></i>;<br/>
<a NAME="E5:76_1_2_2"/><i><font color="Green" title="E72">A15</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  )
 
<b>by </b><i><a class="ref" href="idea_1.html#T44" target="_self" title="IDEA_1:th.44">Th44</a></i>;<br/>
<a NAME="E6:76_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</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> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E14:76_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">M1</font> = <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="idea_1.html#E5:76_1_2_2"><i><font color="Green" title="E72">A15</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E8:76_1_2_2"/><i><font color="Green" title="E73">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:76_1_2_2"><i><font color="Green" title="E70">A13</font></i></a></i>;<br/>
<a NAME="E9:76_1_2_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:76_1_2_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E5:76_1_2_2"><i><font color="Green" title="E72">A15</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:76_1_2_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
 <b>by </b><i><a class="txt" href="idea_1.html#E5:76_1_2_2"><i><font color="Green" title="E72">A15</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</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>
<a NAME="E10:76_1_2_2"/><b>then </b>
<font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E14:76_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="txt" href="idea_1.html#E4:76_1_2_2"><i><font color="Green" title="E71">A14</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E11:76_1_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 <b>by </b><i><a class="txt" href="idea_1.html#E8:76_1_2_2"><i><font color="Green" title="E73">A16</font></i></a>, <a class="ref" href="idea_1.html#T43" target="_self" title="IDEA_1:th.43">Th43</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><a NAME="E17:76_1_2"/>
for <font color="Olive" title="b1">k</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="idea_1.html#E15:76_1_2"><i><font color="Green" title="E68">A10</font></i></a>, <a class="txt" href="idea_1.html#E16:76_1_2"><i><font color="Green" title="E69">A12</font></i></a>);<br/></i><a NAME="E18:76_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">M1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 ;<br/><b>hence </b><a NAME="E19:76_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">M</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="idea_1.html#E10:76_1_2"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="idea_1.html#T43" target="_self" title="IDEA_1:th.43">Th43</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>

</div>
