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<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">lk</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">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#K3" title="SERIES_1:func.3">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="b7">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="b7">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="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/><b>let </b><font color="Maroon" title="c2">lk</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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#K3" title="SERIES_1:func.3">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="b6">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="b6">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">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#K3" title="SERIES_1:func.3">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="b4">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="b4">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K3" title="SERIES_1:func.3">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="b3">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K3" title="SERIES_1:func.3">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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="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/><b>let </b><font color="Maroon" title="c10">m</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="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#K3" title="SERIES_1:func.3">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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="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/>











<b>assume </b><b>that </b><br/><a NAME="E1:78"/><i><font color="Green" title="E60">A1</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:78"/><i><font color="Green" title="E61">A2</font></i>: 
( <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">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 )
 <b>and </b><br/><a NAME="E3:78"/><i><font color="Green" title="E62">A3</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E4:78"/><i><font color="Green" title="E63">A4</font></i>: 
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 )
 <b>and </b><br/><a NAME="E5:78"/><i><font color="Green" title="E64">A5</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E6:78"/><i><font color="Green" title="E65">A6</font></i>: 
( <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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="c8">ke1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 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="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:()"><i><font color="Red">::  thesis: </font></i></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="E7:78"/><i><font color="Green" title="E66">A7</font></i>: 
<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> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E8:78"/><b>then </b>
<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> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><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>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<i><font color="Green" title="E67">A8</font></i>:  <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>
<b>by </b><i><a class="ref" href="idea_1.html#D19" target="_self" title="IDEA_1:def.19">Def19</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">m</font>
<b>by </b><i><a class="ref" href="idea_1.html#D11" target="_self" title="IDEA_1:def.11">Def11</a></i>
;<br/>
<a NAME="E11:78"/>
( <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 &amp; <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 &amp; <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 &amp; <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 )
 
<b>by </b><i><a class="ref" href="idea_1.html#D19" target="_self" title="IDEA_1:def.19">Def19</a></i>;<br/>
<a NAME="E12:78"/><b>then </b><i><font color="Green" title="E68">A9</font></i>: 
( <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> &amp; <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> &amp; <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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="idea_1.html#E2:78"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="idea_1.html#T27" target="_self" title="IDEA_1:th.27">Th27</a></i>;<br/>
<a NAME="E13:78"/><i><font color="Green" title="E69">A10</font></i>: 
<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> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></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"><b>suppose </b></a><a NAME="E1:78_2_1"/><i><font color="Green" title="E70">A11</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">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> 
 <b>by </b><i><a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></i>;<br/><a NAME="E3:78_2_1"/><b>then </b><i><font color="Green" title="E71">A12</font></i>: 
 <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> 
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<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#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> 
 <b>by </b><i><a class="txt" href="idea_1.html#E1:78_2_1"><i><font color="Green" title="E70">A11</font></i></a>, <a class="ref" href="idea_1.html#D18" target="_self" title="IDEA_1:def.18">Def18</a></i>;<br/><b>hence </b><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>
<b>by </b><i><a class="txt" href="idea_1.html#E3:78_2_1"><i><font color="Green" title="E71">A12</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>
<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>
<b>by </b><i><a class="ref" href="funct_7.html#T41" title="FUNCT_7:th.41">FUNCT_7:41</a></i>
<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>
<b>by </b><i><a class="ref" href="funct_7.html#T41" title="FUNCT_7:th.41">FUNCT_7:41</a></i>
<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>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
<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>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
<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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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>
<b>by </b><i><a class="txt" href="idea_1.html#E13:78"><i><font color="Green" title="E69">A10</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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>
<b>by </b><i><a class="txt" href="idea_1.html#E2:78"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="idea_1.html#E6:78"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="idea_1.html#E10:78"><i><font color="Green" title="E67">A8</font></i></a>, <a class="txt" href="idea_1.html#E12:78"><i><font color="Green" title="E68">A9</font></i></a>, <a class="ref" href="idea_1.html#T36" target="_self" title="IDEA_1:th.36">Th36</a></i>
<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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<br/>.= 
<font color="Maroon" title="c10">m</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2:78"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="idea_1.html#E3:78"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="idea_1.html#E5:78"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="idea_1.html#T35" target="_self" title="IDEA_1:th.35">Th35</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_2"><b>suppose </b></a><a NAME="E1:78_2_2"/>
<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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b><i><font color="Green" title="E70">A13</font></i>: 
(  <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>  &amp;  <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>  )
 <b>by </b><i><a class="ref" href="idea_1.html#T41" target="_self" title="IDEA_1:th.41">Th41</a>, <a class="ref" href="idea_1.html#T42" target="_self" title="IDEA_1:th.42">Th42</a></i>;<br/><a NAME="E3:78_2_2"/>
 <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>  is   <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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:78_2_2_1"/><i><font color="Green" title="E71">A14</font></i>: 
(  <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>  &amp;  <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>  )
 
<b>by </b><i><a class="txt" href="idea_1.html#E2:78_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="ref" href="funct_7.html#D9" title="FUNCT_7:def.9">FUNCT_7:def 9</a></i>;<br/>
<a NAME="E2:78_2_2_1"/><b>then </b>
(  <a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">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>  )
 
<b>by </b><i><a class="ref" href="funct_7.html#D6" title="FUNCT_7:def.6">FUNCT_7:def 6</a></i>;<br/>
<a NAME="E3:78_2_2_1"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="funct_7.html#T61" title="FUNCT_7:th.61">FUNCT_7:61</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E4:78_2_2_1"/><b>then </b><i><font color="Green" title="E72">A15</font></i>: 
 <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> 
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:78_2_2_1"><i><font color="Green" title="E71">A14</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:78_2_2_1"/>
 <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> 
 
<b>by </b><i><a class="txt" href="idea_1.html#E2:78_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="idea_1.html#E1:78_2_2_1"><i><font color="Green" title="E71">A14</font></i></a>, <a class="ref" href="funct_7.html#T64" title="FUNCT_7:th.64">FUNCT_7:64</a></i>;<br/>
<b>hence </b><a NAME="E6:78_2_2_1"/>
 <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>  is   <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> 
 <b>by </b><i><a class="txt" href="idea_1.html#E4:78_2_2_1"><i><font color="Green" title="E72">A15</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><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>  ;<br/><a NAME="E5:78_2_2"/>
 <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>  is   <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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:78_2_2_2"/><i><font color="Green" title="E71">A16</font></i>: 
(  <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>  &amp;  <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>  )
 
<b>by </b><i><a class="txt" href="idea_1.html#E2:78_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="ref" href="funct_7.html#D9" title="FUNCT_7:def.9">FUNCT_7:def 9</a></i>;<br/>
<a NAME="E2:78_2_2_2"/><b>then </b>
(  <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>  )
 
<b>by </b><i><a class="ref" href="funct_7.html#D6" title="FUNCT_7:def.6">FUNCT_7:def 6</a></i>;<br/>
<a NAME="E3:78_2_2_2"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="funct_7.html#T61" title="FUNCT_7:th.61">FUNCT_7:61</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E4:78_2_2_2"/><b>then </b><i><font color="Green" title="E72">A17</font></i>: 
 <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> 
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:78_2_2_2"><i><font color="Green" title="E71">A16</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:78_2_2_2"/>
 <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> 
 
<b>by </b><i><a class="txt" href="idea_1.html#E2:78_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="idea_1.html#E1:78_2_2_2"><i><font color="Green" title="E71">A16</font></i></a>, <a class="ref" href="funct_7.html#T64" title="FUNCT_7:th.64">FUNCT_7:64</a></i>;<br/>
<b>hence </b><a NAME="E6:78_2_2_2"/>
 <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>  is   <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> 
 <b>by </b><i><a class="txt" href="idea_1.html#E4:78_2_2_2"><i><font color="Green" title="E72">A17</font></i></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#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c13">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>  ;<br/><a NAME="E7:78_2_2"/><i><font color="Green" title="E71">A18</font></i>: 
(  <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>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">QF</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>  )
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E8:78_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> 
 <b>by </b><i><a class="txt" href="idea_1.html#E13:78"><i><font color="Green" title="E69">A10</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">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>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E10:78_2_2"/><i><font color="Green" title="E72">A19</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">m2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c14">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> &amp; <font color="Maroon" title="c14">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="c14">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> &amp; <font color="Maroon" title="c14">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> )
 <b>by </b><i><a class="txt" href="idea_1.html#E2:78"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="idea_1.html#E10:78"><i><font color="Green" title="E67">A8</font></i></a>, <a class="txt" href="idea_1.html#E12:78"><i><font color="Green" title="E68">A9</font></i></a>, <a class="ref" href="idea_1.html#T47" target="_self" title="IDEA_1:th.47">Th47</a></i>;<br/><a NAME="E11:78_2_2"/><i><font color="Green" title="E73">A20</font></i>: 
<font color="Maroon" title="c14">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> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E12:78_2_2"/><i><font color="Green" title="E74">A21</font></i>: 
<font color="Maroon" title="c13">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_2_3"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon" title="c13">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="c13">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>
<b>by </b><i><a class="txt" href="idea_1.html#E13:78"><i><font color="Green" title="E69">A10</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78_2_2"><i><font color="Green" title="E71">A18</font></i></a>, <a class="ref" href="funct_7.html#T50" title="FUNCT_7:th.50">FUNCT_7:50</a></i>
<br/>.= 
<font color="Maroon" title="c11">m1</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:78"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="idea_1.html#E2:78"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="idea_1.html#E4:78"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="idea_1.html#E10:78"><i><font color="Green" title="E67">A8</font></i></a>, <a class="txt" href="idea_1.html#E12:78"><i><font color="Green" title="E68">A9</font></i></a>, <a class="ref" href="idea_1.html#T48" target="_self" title="IDEA_1:th.48">Th48</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>thus </b><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="c13">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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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="c13">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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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="c13">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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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="c13">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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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="c13">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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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="c13">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="c14">m2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E11:78_2_2"><i><font color="Green" title="E73">A20</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<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>
<b>by </b><i><a class="txt" href="idea_1.html#E2:78"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="idea_1.html#E6:78"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="idea_1.html#E10:78_2_2"><i><font color="Green" title="E72">A19</font></i></a>, <a class="txt" href="idea_1.html#E12:78_2_2"><i><font color="Green" title="E74">A21</font></i></a>, <a class="ref" href="idea_1.html#T36" target="_self" title="IDEA_1:th.36">Th36</a></i>
<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>
<b>by </b><i><a class="txt" href="idea_1.html#E7:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<br/>.= 
<font color="Maroon" title="c10">m</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2:78"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="idea_1.html#E3:78"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="idea_1.html#E5:78"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="idea_1.html#T35" target="_self" title="IDEA_1:th.35">Th35</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
