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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">Key1</font>, <font color="Olive" title="b3">Key2</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="b4">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="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="b1">lk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">r</font> &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &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> &amp; (  for <font color="Olive" title="b6">i</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="Olive" title="b6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">r</font> holds <br/>( <font color="Olive" title="b2">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,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">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,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">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,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">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,4) <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">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,5) <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">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,6) <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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,4) <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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,5) <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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,6) <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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b3">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b3">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b3">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b2">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,5) &amp; <font color="Olive" title="b2">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,6) ) ) holds <br/><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="Olive" title="b2">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b4">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Olive" title="b3">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b4">r</font>)</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="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">m</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Key1</font>, <font color="Olive" title="b2">Key2</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="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> 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="Maroon" title="c2">lk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">r</font> &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &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> &amp; (  for <font color="Olive" title="b5">i</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="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">r</font> holds <br/>( <font color="Olive" title="b1">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,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">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,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">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,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">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,4) <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">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,5) <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">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,6) <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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,4) <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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,5) <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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,6) <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">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b2">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b2">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b2">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b1">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,5) &amp; <font color="Olive" title="b1">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b5">i</font>,6) ) ) holds <br/><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="Olive" title="b1">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Olive" title="b2">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">r</font>)</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="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Key1</font>, <font color="Maroon" title="c4">Key2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> 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="Maroon" title="c2">lk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &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> &amp; (  for <font color="Olive" title="b3">i</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="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> holds <br/>( <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) ) ) holds <br/><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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">r</font>)</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="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font></span><br/>





<a NAME="E1:77"/>
 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> <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>   st <font color="Maroon" title="c2">lk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">r</font> &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  <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> &amp; (  for <font color="Olive" title="b3">i</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="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> holds <br/>( <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) ) ) holds <br/><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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">r</font>)</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="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>   st <font color="Maroon" title="c2">lk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  <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> &amp; (  for <font color="Olive" title="b2">i</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="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> holds <br/>( <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,4) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,5) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,4) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,5) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,5) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,6) ) ) holds <br/><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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">r</font>)</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="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font></span><br/>

<span class="kw">defpred </span><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> ] <span class="kw">means </span>( <font color="Maroon" title="c2">lk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> $1 &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  <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> &amp; (  for <font color="Olive" title="b1">i</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="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/>( <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) ) ) implies <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">Key1</font>,<font color="Maroon" title="c1">n</font>,$1)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,$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="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> );<br/>
<a NAME="E1:77_1"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
 <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">Key1</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>;<br/>
<a NAME="E2:77_1"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
 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="77_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:77_1_1"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">k1</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> <span class="kw"> such that </span><br/><a NAME="E3:77_1_1"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
<font color="Maroon" title="c7">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>
<a NAME="E4:77_1_1"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
 <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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#T43" target="_self" title="IDEA_1:th.43">Th44</a></span>;<br/>
<a NAME="E5:77_1_1"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
 <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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#T43" target="_self" title="IDEA_1:th.43">Th44</a></span>;<br/>
<a NAME="E6:77_1_1"/><span class="lab"><font color="Green" title="E66">A7</font></span>: 
<font color="Maroon" title="c5">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E7:77_1_1"/><span class="kw">then </span>
<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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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="c5">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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">M</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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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="c5">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:77_1_1"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
<font color="Maroon" title="c2">lk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E10:77_1_1"/><span class="lab"><font color="Green" title="E68">A9</font></span>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E11:77_1_1"/><span class="lab"><font color="Green" title="E69">A10</font></span>: 
(  <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 class="kw">and </span><br/><a NAME="E12:77_1_1"/><span class="lab"><font color="Green" title="E70">A11</font></span>: 
 for <font color="Olive" title="b1">i</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="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/>( <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">Key1</font>,<font color="Maroon" title="c1">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font></span><br/>

<a NAME="E13:77_1_1"/><span class="lab"><font color="Green" title="E71">A12</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c8">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77_1_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="ref" href="idea_1.html#T45" target="_self" title="IDEA_1:th.45">Th47</a></span>;<br/>
<a NAME="E14:77_1_1"/><span class="lab"><font color="Green" title="E72">A13</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">Key1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:77_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<a NAME="E15:77_1_1"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">Key1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E16:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A14</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E17:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A15</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a></span>;<br/>
<a NAME="E18:77_1_1"/><span class="lab"><font color="Green" title="E75">A16</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">Key2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:77_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<a NAME="E19:77_1_1"/><span class="kw">then </span>
5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">Key2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E20:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A17</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,5)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E21:77_1_1"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">Key1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:77_1_1"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E22:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A18</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,3)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E23:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A19</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a></span>;<br/>
<a NAME="E24:77_1_1"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">Key1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:77_1_1"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E25:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A20</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,2)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E26:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A21</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a></span>;<br/>
<a NAME="E27:77_1_1"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">Key2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:77_1_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E28:77_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,3)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E29:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A22</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E25:77_1_1"><span class="lab"><font color="Green" title="E79">A20</font></span></a></span>;<br/>
<a NAME="E30:77_1_1"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">Key2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:77_1_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E31:77_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,2)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E32:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A23</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E22:77_1_1"><span class="lab"><font color="Green" title="E77">A18</font></span></a></span>;<br/>
<a NAME="E33:77_1_1"/>
5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">Key1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:77_1_1"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E34:77_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,5)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E35:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A24</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E20:77_1_1"><span class="lab"><font color="Green" title="E76">A17</font></span></a></span>;<br/>
<a NAME="E36:77_1_1"/><span class="lab"><font color="Green" title="E84">A25</font></span>: 
<font color="Maroon" title="c8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77_1_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="ref" href="idea_1.html#T45" target="_self" title="IDEA_1:th.45">Th47</a></span>;<br/>
<a NAME="E37:77_1_1"/><span class="lab"><font color="Green" title="E85">A26</font></span>: 
( <font color="Maroon" title="c8">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="c8">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77_1_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="ref" href="idea_1.html#T45" target="_self" title="IDEA_1:th.45">Th47</a></span>;<br/>
<a NAME="E38:77_1_1"/>
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">Key1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:77_1_1"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E39:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A27</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,4)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E40:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A28</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a></span>;<br/>
<a NAME="E41:77_1_1"/>
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">Key2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:77_1_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E42:77_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,4)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E43:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A29</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E39:77_1_1"><span class="lab"><font color="Green" title="E86">A27</font></span></a></span>;<br/>
<a NAME="E44:77_1_1"/>
6 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">Key2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:77_1_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E45:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A30</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,6)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E46:77_1_1"/>
6 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">Key1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:77_1_1"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E47:77_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,6)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E48:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A31</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E45:77_1_1"><span class="lab"><font color="Green" title="E89">A30</font></span></a></span>;<br/>
<a NAME="E49:77_1_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#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <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">Key1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E50:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A32</font></span>: 
 <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"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">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>)</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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E4:77_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
 <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"><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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span> = 
 <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="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#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c6">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">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"><span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T41" title="FUNCT_7:th.41">FUNCT_7:41</a></span>
;<br/>
<a NAME="E52:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A33</font></span>: 
 <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"><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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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></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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E53:77_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">Key2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:77_1_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E54:77_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k1</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E55:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A34</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:77_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E16:77_1_1"><span class="lab"><font color="Green" title="E73">A14</font></span></a></span>;<br/>
<a NAME="E56:77_1_1"/><span class="lab"><font color="Green" title="E94">A35</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E57:77_1_1"/><span class="lab"><font color="Green" title="E95">A36</font></span>: 
 for <font color="Olive" title="b1">i</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="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> holds <br/>( <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,5) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,6) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> implies ( <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:77_1_1_2"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) )</span><br/>

<a NAME="E2:77_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:77_1_1"><span class="lab"><font color="Green" title="E94">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:77_1_1_2"/>
( <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,5) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i</font>,6) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:77_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<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">Key1</font>,<font color="Maroon" title="c1">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c5">m</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#T36" target="_self" title="IDEA_1:th.36">Th37</a></span>
<br/>.= 
<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"><span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span></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="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#T37" target="_self" title="IDEA_1:th.37">Th38</a></span>
<br/>.= 
<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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p5"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</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="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>
<br/>.= 
<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"><span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">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> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p5"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">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></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</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="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>
<br/>.= 
<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#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</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"><span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">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> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p5"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">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></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="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:77_1_1"><span class="lab"><font color="Green" title="E92">A33</font></span></a>, <a class="ref" href="funct_7.html#T48" title="FUNCT_7:th.48">FUNCT_7:48</a></span>
<br/>.= 
<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#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">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> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p5"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">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#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="funct_7.html#T48" title="FUNCT_7:th.48">FUNCT_7:48</a></span>
<br/>.= 
<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#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p4"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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 class="p5">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></span>
<br/>.= 
<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#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<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></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T51" title="FUNCT_7:th.51">FUNCT_7:51</a></span>
<br/>.= 
<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#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c6">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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <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">Key1</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c6">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>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>
<br/>.= 
<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#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c6">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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <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">Key1</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c6">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>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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="c5">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:77_1_1"><span class="lab"><font color="Green" title="E91">A32</font></span></a>, <a class="txt" href="#E6:77_1_1"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<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#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c4">Key2</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> (<span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">Key1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">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>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p4">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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="c5">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E6:77_1_1"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<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#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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> <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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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="c5">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:77_1_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="txt" href="#E13:77_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E37:77_1_1"><span class="lab"><font color="Green" title="E85">A26</font></span></a>, <a class="txt" href="#E36:77_1_1"><span class="lab"><font color="Green" title="E84">A25</font></span></a>, <a class="txt" href="#E17:77_1_1"><span class="lab"><font color="Green" title="E74">A15</font></span></a>, <a class="txt" href="#E26:77_1_1"><span class="lab"><font color="Green" title="E80">A21</font></span></a>, <a class="txt" href="#E23:77_1_1"><span class="lab"><font color="Green" title="E78">A19</font></span></a>, <a class="txt" href="#E40:77_1_1"><span class="lab"><font color="Green" title="E87">A28</font></span></a>, <a class="txt" href="#E55:77_1_1"><span class="lab"><font color="Green" title="E93">A34</font></span></a>, <a class="txt" href="#E32:77_1_1"><span class="lab"><font color="Green" title="E82">A23</font></span></a>, <a class="txt" href="#E29:77_1_1"><span class="lab"><font color="Green" title="E81">A22</font></span></a>, <a class="txt" href="#E43:77_1_1"><span class="lab"><font color="Green" title="E88">A29</font></span></a>, <a class="txt" href="#E35:77_1_1"><span class="lab"><font color="Green" title="E83">A24</font></span></a>, <a class="txt" href="#E48:77_1_1"><span class="lab"><font color="Green" title="E90">A31</font></span></a>, <a class="ref" href="idea_1.html#T33" target="_self" title="IDEA_1:th.33">Th34</a></span>
<br/>.= 
<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#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">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="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E6:77_1_1"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E9:77_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E10:77_1_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="txt" href="#E11:77_1_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E56:77_1_1"><span class="lab"><font color="Green" title="E94">A35</font></span></a>, <a class="txt" href="#E57:77_1_1"><span class="lab"><font color="Green" title="E95">A36</font></span></a>, <a class="txt" href="#E5:77_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="funct_7.html#T48" title="FUNCT_7:th.48">FUNCT_7:48</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E59:77_1_1"/>
<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">Key1</font>,<font color="Maroon" title="c1">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:77_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#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D18" target="_self" title="IDEA_1:def.18">Def18</a></span>;<br/>
<a NAME="E4:77_1"/><span class="kw">then </span>
 <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</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> 
 
;<br/>
<span class="kw">then </span><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">Key1</font>,<font color="Maroon" title="c1">n</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</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#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E60">A1</font></span></a></span>
;<br/>
<a NAME="E6:77_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">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>  &amp;  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><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">Key1</font>,<font color="Maroon" title="c1">n</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span></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>  <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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a>, <a class="ref" href="funct_7.html#T39" title="FUNCT_7:th.39">FUNCT_7:39</a></span>;<br/>
<a NAME="E7:77_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A37</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:77_1"/>
 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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E7:77_1"><span class="lab"><font color="Green" title="E62">A37</font></span></a>, <a class="txt" href="#E2:77_1"><span class="lab"><font color="Green" title="E61">A2</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:77"/>
 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> 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="Maroon" title="c2">lk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &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> &amp; (  for <font color="Olive" title="b3">i</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="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> holds <br/>( <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) <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="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) <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="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,1)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,3)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,2)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,4)</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,5) &amp; <font color="Maroon" title="c3">Key1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i</font>,6) ) ) holds <br/><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">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c4">Key2</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">r</font>)</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="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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