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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">ks1</font>, <font color="Olive" title="b6">ks2</font>, <font color="Olive" title="b7">ke1</font>, <font color="Olive" title="b8">ke2</font>, <font color="Olive" title="b9">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="b9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b9">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="b9">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="b9">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="b9">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">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="b10">i</font>,6) ) ) &amp; <font color="Olive" title="b5">ks1</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">ks1</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">ks1</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">ks1</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; <font color="Olive" title="b6">ks2</font> <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"><font color="Olive" title="b5">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b6">ks2</font> <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"><font color="Olive" title="b5">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b6">ks2</font> <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"><font color="Olive" title="b5">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b6">ks2</font> <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"><font color="Olive" title="b5">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b7">ke1</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="b7">ke1</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="b7">ke1</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="b7">ke1</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; <font color="Olive" title="b8">ke2</font> <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"><font color="Olive" title="b7">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b8">ke2</font> <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"><font color="Olive" title="b7">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b8">ke2</font> <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"><font color="Olive" title="b7">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b8">ke2</font> <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"><font color="Olive" title="b7">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b8">ke2</font> <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="Olive" title="b7">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 &amp; <font color="Olive" title="b8">ke2</font> <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="Olive" title="b7">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Olive" title="b6">ks2</font>,<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"><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#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>,<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"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Olive" title="b8">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Olive" title="b7">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Olive" title="b2">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b4">r</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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Olive" title="b5">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">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">ks1</font>, <font color="Olive" title="b5">ks2</font>, <font color="Olive" title="b6">ke1</font>, <font color="Olive" title="b7">ke2</font>, <font color="Olive" title="b8">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="b8">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b8">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="b8">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="b8">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="b8">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">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="b9">i</font>,6) ) ) &amp; <font color="Olive" title="b4">ks1</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">ks1</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">ks1</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">ks1</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; <font color="Olive" title="b5">ks2</font> <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"><font color="Olive" title="b4">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b5">ks2</font> <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"><font color="Olive" title="b4">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b5">ks2</font> <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"><font color="Olive" title="b4">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b5">ks2</font> <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"><font color="Olive" title="b4">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b6">ke1</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="b6">ke1</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="b6">ke1</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="b6">ke1</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; <font color="Olive" title="b7">ke2</font> <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"><font color="Olive" title="b6">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b7">ke2</font> <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"><font color="Olive" title="b6">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b7">ke2</font> <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"><font color="Olive" title="b6">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b7">ke2</font> <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"><font color="Olive" title="b6">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b7">ke2</font> <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="Olive" title="b6">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 &amp; <font color="Olive" title="b7">ke2</font> <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="Olive" title="b6">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Olive" title="b5">ks2</font>,<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"><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#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>,<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"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Olive" title="b7">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Olive" title="b6">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Olive" title="b1">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">r</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Olive" title="b4">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">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">ks1</font>, <font color="Olive" title="b3">ks2</font>, <font color="Olive" title="b4">ke1</font>, <font color="Olive" title="b5">ke2</font>, <font color="Olive" title="b6">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="b6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Olive" title="b6">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="b6">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="b6">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="b6">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">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="b7">i</font>,6) ) ) &amp; <font color="Olive" title="b2">ks1</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">ks1</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">ks1</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">ks1</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; <font color="Olive" title="b3">ks2</font> <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"><font color="Olive" title="b2">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b3">ks2</font> <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"><font color="Olive" title="b2">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b3">ks2</font> <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"><font color="Olive" title="b2">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b3">ks2</font> <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"><font color="Olive" title="b2">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b4">ke1</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">ke1</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">ke1</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">ke1</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; <font color="Olive" title="b5">ke2</font> <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"><font color="Olive" title="b4">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b5">ke2</font> <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"><font color="Olive" title="b4">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b5">ke2</font> <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"><font color="Olive" title="b4">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b5">ke2</font> <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"><font color="Olive" title="b4">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b5">ke2</font> <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="Olive" title="b4">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 &amp; <font color="Olive" title="b5">ke2</font> <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="Olive" title="b4">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Olive" title="b3">ks2</font>,<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"><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#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>,<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"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Olive" title="b5">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Olive" title="b4">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">r</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Olive" title="b2">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">ks1</font>, <font color="Olive" title="b2">ks2</font>, <font color="Olive" title="b3">ke1</font>, <font color="Olive" title="b4">ke2</font>, <font color="Olive" title="b5">m</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">lk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">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="Maroon" title="c5">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="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="Maroon" title="c3">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="Maroon" title="c3">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="Maroon" title="c3">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="Maroon" title="c3">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="Maroon" title="c3">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="Maroon" title="c4">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="Maroon" title="c4">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="Maroon" title="c4">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="Maroon" title="c4">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="Maroon" title="c4">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="Maroon" title="c4">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="Maroon" title="c4">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="Maroon" title="c3">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="Maroon" title="c4">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="Maroon" title="c3">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="Maroon" title="c4">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="Maroon" title="c3">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="Maroon" title="c4">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="Maroon" title="c3">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="Maroon" title="c3">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="Maroon" title="c4">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="Maroon" title="c3">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="Maroon" title="c4">Key2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b6">i</font>,6) ) ) &amp; <font color="Olive" title="b1">ks1</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">ks1</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">ks1</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">ks1</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; <font color="Olive" title="b2">ks2</font> <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"><font color="Olive" title="b1">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b2">ks2</font> <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"><font color="Olive" title="b1">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b2">ks2</font> <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"><font color="Olive" title="b1">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b2">ks2</font> <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"><font color="Olive" title="b1">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b3">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">ke1</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; <font color="Olive" title="b4">ke2</font> <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"><font color="Olive" title="b3">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b4">ke2</font> <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"><font color="Olive" title="b3">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b4">ke2</font> <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"><font color="Olive" title="b3">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b4">ke2</font> <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"><font color="Olive" title="b3">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b4">ke2</font> <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="Olive" title="b3">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 &amp; <font color="Olive" title="b4">ke2</font> <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="Olive" title="b3">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Olive" title="b2">ks2</font>,<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"><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#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="c5">r</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Olive" title="b4">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Olive" title="b3">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">r</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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Olive" title="b1">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></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="c6">ks1</font>, <font color="Maroon" title="c7">ks2</font>, <font color="Maroon" title="c8">ke1</font>, <font color="Maroon" title="c9">ke2</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">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="Maroon" title="c5">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="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="Maroon" title="c5">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) ) ) &amp; <font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">ks1</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; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c8">ke1</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="c8">ke1</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">ke1</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="c8">ke1</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; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 &amp; <font color="Maroon" title="c9">ke2</font> <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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><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#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="c5">r</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">r</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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></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></span><br/><span class="kw">let </span><font color="Maroon" title="c10">m</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">lk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">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="c10">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c10">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="c10">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="c10">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="c10">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> <font color="Maroon" title="c5">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="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) ) ) &amp; <font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">ks1</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; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c8">ke1</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="c8">ke1</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">ke1</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="c8">ke1</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; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 &amp; <font color="Maroon" title="c9">ke2</font> <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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><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#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="c5">r</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">r</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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:78"/><span class="lab"><font color="Green" title="E60">A1</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="c5">r</font>
 <span class="kw">and </span><br/><a NAME="E2:78"/><span class="lab"><font color="Green" title="E61">A2</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="E3:78"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 <span class="kw">and </span><br/><a NAME="E4:78"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
( <font color="Maroon" title="c10">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="c10">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="c10">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="c10">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="E5:78"/><span class="lab"><font color="Green" title="E64">A5</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="c5">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="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) )
 <span class="kw">and </span><br/><a NAME="E6:78"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
( <font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">ks1</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; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c7">ks2</font> <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"><font color="Maroon" title="c6">ks1</font> <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">and </span><br/><a NAME="E7:78"/><span class="lab"><font color="Green" title="E66">A7</font></span>: 
( <font color="Maroon" title="c8">ke1</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="c8">ke1</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">ke1</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="c8">ke1</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; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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"><font color="Maroon" title="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c9">ke2</font> <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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 &amp; <font color="Maroon" title="c9">ke2</font> <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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><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#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="c5">r</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">r</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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font></span><br/>

<a NAME="E8:78"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
<font color="Maroon" title="c10">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="E9:78"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">m1</font> = <span class="p1">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E11:78"/><span class="lab"><font color="Green" title="E68">A9</font></span>: 
<font color="Maroon" title="c11">m1</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/>
<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="c11">m1</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<font color="Maroon" title="c10">m</font>,<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D19" target="_self" title="IDEA_1:def.19">Def19</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D11" target="_self" title="IDEA_1:def.11">Def11</a></span>
;<br/>
<a NAME="E13:78"/>
<font color="Maroon" title="c11">m1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<font color="Maroon" title="c10">m</font>,<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D19" target="_self" title="IDEA_1:def.19">Def19</a></span>;<br/>
<a NAME="E14:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A11</font></span>: 
<font color="Maroon" title="c11">m1</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="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="idea_1.html#T26" target="_self" title="IDEA_1:th.26">Th27</a></span>;<br/>
<a NAME="E15:78"/>
<font color="Maroon" title="c11">m1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<font color="Maroon" title="c10">m</font>,<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D19" target="_self" title="IDEA_1:def.19">Def19</a></span>;<br/>
<a NAME="E16:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A12</font></span>: 
<font color="Maroon" title="c11">m1</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="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="idea_1.html#T26" target="_self" title="IDEA_1:th.26">Th27</a></span>;<br/>
<a NAME="E17:78"/>
<font color="Maroon" title="c11">m1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<font color="Maroon" title="c10">m</font>,<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D19" target="_self" title="IDEA_1:def.19">Def19</a></span>;<br/>
<a NAME="E18:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A13</font></span>: 
<font color="Maroon" title="c11">m1</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="idea_1.html#T26" target="_self" title="IDEA_1:th.26">Th27</a></span>;<br/>
<a NAME="E19:78"/>
<font color="Maroon" title="c11">m1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<font color="Maroon" title="c10">m</font>,<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D19" target="_self" title="IDEA_1:def.19">Def19</a></span>;<br/>
<a NAME="E20:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A14</font></span>: 
<font color="Maroon" title="c11">m1</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="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="idea_1.html#T26" target="_self" title="IDEA_1:th.26">Th27</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:78_2"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1"><span class="kw">suppose </span></a><a NAME="E1:78_2_1"/><span class="lab"><font color="Green" title="E74">A15</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><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#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="c5">r</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">r</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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font></span><br/><div class="add"><a NAME="E2:78_2_1"/><span class="kw">then </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#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="c5">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <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="E3:78_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A16</font></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>,<font color="Maroon" title="c5">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E4:78_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_2_1"><span class="lab"><font color="Green" title="E74">A15</font></span></a>, <a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>;<br/><a NAME="E5:78_2_1"/><span class="kw">then </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>,<font color="Maroon" title="c5">r</font>) <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">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><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#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="c5">r</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">r</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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><span class="p3">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<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"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_2_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T39" title="FUNCT_7:th.39">FUNCT_7:39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<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"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<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"><span class="p0">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T39" title="FUNCT_7:th.39">FUNCT_7:39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><span class="p3">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<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"><span class="p4">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<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"><span class="p5">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><span class="p3">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<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"><span class="p4">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<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#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<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"><span class="p4">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<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#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</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"><span class="p4">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<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#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</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="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<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#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:78"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E7:78"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E20:78"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="txt" href="#E18:78"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="txt" href="#E16:78"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E14:78"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="idea_1.html#T35" target="_self" title="IDEA_1:th.35">Th36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="idea_1.html#T34" target="_self" title="IDEA_1:th.34">Th35</a></span>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_2"><span class="kw">suppose </span></a><a NAME="E1:78_2_2"/><span class="lab"><font color="Green" title="E74">A17</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><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#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="c5">r</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">r</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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font></span><br/><div class="add"><a NAME="E2:78_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A18</font></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>,<font color="Maroon" title="c5">r</font>) is    <a href="funct_7.html#M1" title="FUNCT_7:mode.1">FuncSequence</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</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#T40" target="_self" title="IDEA_1:th.40">Th41</a></span>;<br/><a NAME="E3:78_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A19</font></span>: 
 <a href="funct_7.html#K6" title="FUNCT_7:func.6">firstdom</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>,<font color="Maroon" title="c5">r</font>)</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_7.html#D9" title="FUNCT_7:def.9">FUNCT_7:def 9</a></span>;<br/><a NAME="E4:78_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A20</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="c5">r</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="txt" href="#E2:78_2_2"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="ref" href="funct_7.html#T62" title="FUNCT_7:th.62">FUNCT_7:62</a></span>;<br/><a NAME="E5:78_2_2"/>
(  <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="c5">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <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="c5">r</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_2_2"><span class="lab"><font color="Green" title="E76">A19</font></span></a>, <a class="ref" href="funct_7.html#D6" title="FUNCT_7:def.6">FUNCT_7:def 6</a></span>;<br/><a NAME="E6:78_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A21</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="c5">r</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="funct_7.html#K7" title="FUNCT_7:func.7">lastrng</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>,<font color="Maroon" title="c5">r</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T59" title="FUNCT_7:th.59">FUNCT_7:59</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/><a NAME="E7:78_2_2"/>
 <a href="funct_7.html#K7" title="FUNCT_7:func.7">lastrng</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>,<font color="Maroon" title="c5">r</font>)</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="txt" href="#E2:78_2_2"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="ref" href="funct_7.html#D9" title="FUNCT_7:def.9">FUNCT_7:def 9</a></span>;<br/><a NAME="E8:78_2_2"/><span class="kw">then </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="c5">r</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="txt" href="#E6:78_2_2"><span class="lab"><font color="Green" title="E78">A21</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">PF</font> =  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">r</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>) as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</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="#E4:78_2_2"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/><a NAME="E10:78_2_2"/><span class="lab"><font color="Green" title="E79">A22</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">PF</font> <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="E11:78_2_2"/>
<font color="Maroon" title="c12">PF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m1</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="txt" href="#E11:78"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">m2</font> = <font color="Maroon" title="c12">PF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m1</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="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><a NAME="E13:78_2_2"/><span class="lab"><font color="Green" title="E80">A23</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">m2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c13">m2</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="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E20:78"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="txt" href="#E18:78"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="txt" href="#E16:78"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E14:78"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="idea_1.html#T45" target="_self" title="IDEA_1:th.45">Th47</a></span>;<br/><a NAME="E14:78_2_2"/><span class="lab"><font color="Green" title="E81">A24</font></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>,<font color="Maroon" title="c5">r</font>) is    <a href="funct_7.html#M1" title="FUNCT_7:mode.1">FuncSequence</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</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="#E1:78_2_2"><span class="lab"><font color="Green" title="E74">A17</font></span></a>, <a class="ref" href="idea_1.html#T41" target="_self" title="IDEA_1:th.41">Th42</a></span>;<br/><a NAME="E15:78_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A25</font></span>: 
 <a href="funct_7.html#K6" title="FUNCT_7:func.6">firstdom</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>,<font color="Maroon" title="c5">r</font>)</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_7.html#D9" title="FUNCT_7:def.9">FUNCT_7:def 9</a></span>;<br/><a NAME="E16:78_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A26</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#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="c5">r</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="txt" href="#E14:78_2_2"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="ref" href="funct_7.html#T62" title="FUNCT_7:th.62">FUNCT_7:62</a></span>;<br/><a NAME="E17:78_2_2"/>
(  <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="c5">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <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#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="c5">r</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:78_2_2"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="ref" href="funct_7.html#D6" title="FUNCT_7:def.6">FUNCT_7:def 6</a></span>;<br/><a NAME="E18:78_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A27</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#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="c5">r</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="funct_7.html#K7" title="FUNCT_7:func.7">lastrng</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>,<font color="Maroon" title="c5">r</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T59" title="FUNCT_7:th.59">FUNCT_7:59</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/><a NAME="E19:78_2_2"/>
 <a href="funct_7.html#K7" title="FUNCT_7:func.7">lastrng</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>,<font color="Maroon" title="c5">r</font>)</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="txt" href="#E14:78_2_2"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="ref" href="funct_7.html#D9" title="FUNCT_7:def.9">FUNCT_7:def 9</a></span>;<br/><a NAME="E20:78_2_2"/><span class="kw">then </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#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="c5">r</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="txt" href="#E18:78_2_2"><span class="lab"><font color="Green" title="E84">A27</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">QF</font> =  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</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>,<font color="Maroon" title="c5">r</font>)</span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>) as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</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="#E16:78_2_2"><span class="lab"><font color="Green" title="E83">A26</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/><a NAME="E22:78_2_2"/><span class="lab"><font color="Green" title="E85">A28</font></span>: 
<font color="Maroon" title="c13">m2</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/><span class="lab"><font color="Green" title="E86">A29</font></span>: <font color="Maroon" title="c14">QF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">PF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">QF</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">PF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:78"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</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"><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="c5">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="Maroon" title="c5">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="c11">m1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:78_2_2"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="funct_7.html#T48" title="FUNCT_7:th.48">FUNCT_7:48</a></span>
<br/>.= 
<font color="Maroon" title="c11">m1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E5:78"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E20:78"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="txt" href="#E18:78"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="txt" href="#E16:78"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E14:78"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="idea_1.html#T46" target="_self" title="IDEA_1:th.46">Th48</a></span>
;<br/><a NAME="E24:78_2_2"/><span class="lab"><font color="Green" title="E87">A30</font></span>: 
<font color="Maroon" title="c13">m2</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="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E20:78"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="txt" href="#E18:78"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="txt" href="#E16:78"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E14:78"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="idea_1.html#T45" target="_self" title="IDEA_1:th.45">Th47</a></span>;<br/><a NAME="E25:78_2_2"/><span class="lab"><font color="Green" title="E88">A31</font></span>: 
( <font color="Maroon" title="c13">m2</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="c13">m2</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="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E20:78"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="txt" href="#E18:78"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="txt" href="#E16:78"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E14:78"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="idea_1.html#T45" target="_self" title="IDEA_1:th.45">Th47</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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"><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#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="c5">r</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> (<font color="Maroon" title="c3">Key1</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">r</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="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> = 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">QF</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<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"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<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"><font color="Maroon" title="c12">PF</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">QF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<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"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<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"><font color="Maroon" title="c12">PF</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">QF</font> <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="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<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"><font color="Maroon" title="c12">PF</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">QF</font> <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="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">PF</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">QF</font> <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="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">PF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">QF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K22" title="IDEA_1:func.22">IDEA_QE</a> (<font color="Maroon" title="c9">ke2</font>,<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#K21" title="IDEA_1:func.21">IDEA_PE</a> (<font color="Maroon" title="c8">ke1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:78_2_2"><span class="lab"><font color="Green" title="E85">A28</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E7:78"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E13:78_2_2"><span class="lab"><font color="Green" title="E80">A23</font></span></a>, <a class="txt" href="#E25:78_2_2"><span class="lab"><font color="Green" title="E88">A31</font></span></a>, <a class="txt" href="#E24:78_2_2"><span class="lab"><font color="Green" title="E87">A30</font></span></a>, <a class="txt" href="#E23:78_2_2"><span class="lab"><font color="Green" title="E86">A29</font></span></a>, <a class="ref" href="idea_1.html#T35" target="_self" title="IDEA_1:th.35">Th36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K20" title="IDEA_1:func.20">IDEA_QS</a> (<font color="Maroon" title="c7">ks2</font>,<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#K19" title="IDEA_1:func.19">IDEA_PS</a> (<font color="Maroon" title="c6">ks1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="idea_1.html#T34" target="_self" title="IDEA_1:th.34">Th35</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></div><span class="kw">end;</span></div>

</div>
