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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> &amp; (  for <font color="Olive" title="b4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> ) &amp; (  for <font color="Olive" title="b4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">g</font> holds <br/><font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> ) holds <br/> ex <font color="Olive" title="b4">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Olive" title="b2">f</font></span>)</span></span>)</span> st  <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">b</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> &amp; (  for <font color="Olive" title="b3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> ) &amp; (  for <font color="Olive" title="b3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">g</font> holds <br/><font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> ) holds <br/> ex <font color="Olive" title="b3">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> st  <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> ) &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> ) implies  ex <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> st  <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">p</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:162"/><span class="lab"><font color="Green" title="E73">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:162"/><span class="lab"><font color="Green" title="E74">A2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:162"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:162"/><span class="lab"><font color="Green" title="E76">A4</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> st  <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">p</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">Ff</font> =  <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">Fg</font> =  <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">db</font> =  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font>;<br/>
<a NAME="E5:162"/><span class="lab"><font color="Green" title="E77">A5</font></span>: 
 <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="162_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k1</font>, <font color="Maroon" title="c9">k2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> or  not <font color="Maroon" title="c9">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k2</font> or <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:162_1"/><span class="lab"><font color="Green" title="E78">A6</font></span>: 
<font color="Maroon" title="c8">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:162_1"/><span class="lab"><font color="Green" title="E79">A7</font></span>: 
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:162_1"/><span class="lab"><font color="Green" title="E80">A8</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">i1</font>, <font color="Maroon" title="c11">j1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:162_1"/><span class="lab"><font color="Green" title="E81">A9</font></span>: 
<font color="Maroon" title="c10">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">and </span><br/><a NAME="E6:162_1"/><span class="lab"><font color="Green" title="E82">A10</font></span>: 
<font color="Maroon" title="c11">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:162_1"/><span class="lab"><font color="Green" title="E83">A11</font></span>: 
<font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">j1</font>
 <span class="kw">and </span><br/><a NAME="E8:162_1"/><span class="lab"><font color="Green" title="E84">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162_1"><span class="lab"><font color="Green" title="E78">A6</font></span></a>, <a class="ref" href="pre_poly.html#T29" target="_self" title="PRE_POLY:th.29">Th29</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">dbi1</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c10">i1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">ddbi11</font> =  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>;<br/>
<a NAME="E10:162_1"/><span class="lab"><font color="Green" title="E85">A13</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:162"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="txt" href="#E5:162_1"><span class="lab"><font color="Green" title="E81">A9</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E86">A14</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>

;<br/>
<a NAME="E12:162_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A15</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162_1"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E13:162_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E14:162_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A16</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162_1"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="txt" href="#E11:162_1"><span class="lab"><font color="Green" title="E86">A14</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">b11</font>, <font color="Maroon" title="c15">b12</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E16:162_1"/><span class="lab"><font color="Green" title="E89">A17</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c14">b11</font>,<font color="Maroon" title="c15">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E17:162_1"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b11</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c15">b12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:162"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E5:162_1"><span class="lab"><font color="Green" title="E81">A9</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">b119</font> = <font color="Maroon" title="c14">b11</font>, <font color="Maroon" title="c17">b129</font> = <font color="Maroon" title="c15">b12</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" target="_self" title="PRE_POLY:def.12">Def12</a></span>;<br/>
<a NAME="E19:162_1"/><span class="lab"><font color="Green" title="E90">A18</font></span>: 
( <font color="Maroon" title="c16">b119</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b11</font> &amp; <font color="Maroon" title="c17">b129</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b12</font> )
 
;<br/>
<a NAME="E20:162_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:162_1"><span class="lab"><font color="Green" title="E89">A17</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">b111</font>, <font color="Maroon" title="c19">b112</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E22:162_1"/><span class="lab"><font color="Green" title="E91">A19</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c18">b111</font>,<font color="Maroon" title="c19">b112</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E23:162_1"/><span class="lab"><font color="Green" title="E92">A20</font></span>: 
<font color="Maroon" title="c15">b12</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b111</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b112</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162_1"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="txt" href="#E11:162_1"><span class="lab"><font color="Green" title="E86">A14</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/>
<a NAME="E24:162_1"/>
<font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:162_1"><span class="lab"><font color="Green" title="E89">A17</font></span></a>, <a class="txt" href="#E19:162_1"><span class="lab"><font color="Green" title="E90">A18</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E93">A21</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j1</font> = 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c14">b11</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c18">b111</font>,<font color="Maroon" title="c19">b112</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162_1"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="txt" href="#E10:162_1"><span class="lab"><font color="Green" title="E85">A13</font></span></a>, <a class="txt" href="#E22:162_1"><span class="lab"><font color="Green" title="E91">A19</font></span></a>, <a class="txt" href="#E12:162_1"><span class="lab"><font color="Green" title="E87">A15</font></span></a>, <a class="txt" href="#E14:162_1"><span class="lab"><font color="Green" title="E88">A16</font></span></a>, <a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c14">b11</font>,<font color="Maroon" title="c18">b111</font>,<font color="Maroon" title="c19">b112</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">i2</font>, <font color="Maroon" title="c21">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E27:162_1"/><span class="lab"><font color="Green" title="E94">A22</font></span>: 
<font color="Maroon" title="c20">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">and </span><br/><a NAME="E28:162_1"/><span class="lab"><font color="Green" title="E95">A23</font></span>: 
<font color="Maroon" title="c21">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E29:162_1"/><span class="lab"><font color="Green" title="E96">A24</font></span>: 
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">j2</font>
 <span class="kw">and </span><br/><a NAME="E30:162_1"/><span class="lab"><font color="Green" title="E97">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:162_1"><span class="lab"><font color="Green" title="E79">A7</font></span></a>, <a class="ref" href="pre_poly.html#T29" target="_self" title="PRE_POLY:th.29">Th29</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">dbi2</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c20">i2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">ddbi21</font> =  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>;<br/>
<a NAME="E32:162_1"/><span class="lab"><font color="Green" title="E98">A26</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:162"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="txt" href="#E27:162_1"><span class="lab"><font color="Green" title="E94">A22</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E99">A27</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p5">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>

;<br/>
<a NAME="E34:162_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A28</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:162_1"><span class="lab"><font color="Green" title="E95">A23</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E35:162_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E36:162_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A29</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:162_1"><span class="lab"><font color="Green" title="E95">A23</font></span></a>, <a class="txt" href="#E33:162_1"><span class="lab"><font color="Green" title="E99">A27</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">b21</font>, <font color="Maroon" title="c25">b22</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E38:162_1"/><span class="lab"><font color="Green" title="E102">A30</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c24">b21</font>,<font color="Maroon" title="c25">b22</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E39:162_1"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b21</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c25">b22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:162"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E27:162_1"><span class="lab"><font color="Green" title="E94">A22</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">b219</font> = <font color="Maroon" title="c24">b21</font>, <font color="Maroon" title="c27">b229</font> = <font color="Maroon" title="c25">b22</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" target="_self" title="PRE_POLY:def.12">Def12</a></span>;<br/>
<a NAME="E41:162_1"/><span class="lab"><font color="Green" title="E103">A31</font></span>: 
( <font color="Maroon" title="c26">b219</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b21</font> &amp; <font color="Maroon" title="c27">b229</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">b22</font> )
 
;<br/>
<a NAME="E42:162_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">b22</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:162_1"><span class="lab"><font color="Green" title="E102">A30</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">b211</font>, <font color="Maroon" title="c29">b212</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E44:162_1"/><span class="lab"><font color="Green" title="E104">A32</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c28">b211</font>,<font color="Maroon" title="c29">b212</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E45:162_1"/><span class="lab"><font color="Green" title="E105">A33</font></span>: 
<font color="Maroon" title="c25">b22</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c28">b211</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c29">b212</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:162_1"><span class="lab"><font color="Green" title="E95">A23</font></span></a>, <a class="txt" href="#E33:162_1"><span class="lab"><font color="Green" title="E99">A27</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/>
<a NAME="E46:162_1"/>
<font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b21</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:162_1"><span class="lab"><font color="Green" title="E102">A30</font></span></a>, <a class="txt" href="#E41:162_1"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E106">A34</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j2</font> = 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c24">b21</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c28">b211</font>,<font color="Maroon" title="c29">b212</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:162_1"><span class="lab"><font color="Green" title="E95">A23</font></span></a>, <a class="txt" href="#E32:162_1"><span class="lab"><font color="Green" title="E98">A26</font></span></a>, <a class="txt" href="#E44:162_1"><span class="lab"><font color="Green" title="E104">A32</font></span></a>, <a class="txt" href="#E34:162_1"><span class="lab"><font color="Green" title="E100">A28</font></span></a>, <a class="txt" href="#E36:162_1"><span class="lab"><font color="Green" title="E101">A29</font></span></a>, <a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c24">b21</font>,<font color="Maroon" title="c28">b211</font>,<font color="Maroon" title="c29">b212</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>
;<br/>
<a NAME="E48:162_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A35</font></span>: 
( <font color="Maroon" title="c18">b111</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c28">b211</font> &amp; <font color="Maroon" title="c19">b112</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c29">b212</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162_1"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="txt" href="#E8:162_1"><span class="lab"><font color="Green" title="E84">A12</font></span></a>, <a class="txt" href="#E30:162_1"><span class="lab"><font color="Green" title="E97">A25</font></span></a>, <a class="txt" href="#E25:162_1"><span class="lab"><font color="Green" title="E93">A21</font></span></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></span>;<br/>
<a NAME="E49:162_1"/><span class="lab"><font color="Green" title="E108">A36</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:162"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E27:162_1"><span class="lab"><font color="Green" title="E94">A22</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E50:162_1"/><span class="lab"><font color="Green" title="E109">A37</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:162"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E5:162_1"><span class="lab"><font color="Green" title="E81">A9</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E51:162_1"/>
<font color="Maroon" title="c14">b11</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b21</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162_1"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="txt" href="#E8:162_1"><span class="lab"><font color="Green" title="E84">A12</font></span></a>, <a class="txt" href="#E30:162_1"><span class="lab"><font color="Green" title="E97">A25</font></span></a>, <a class="txt" href="#E25:162_1"><span class="lab"><font color="Green" title="E93">A21</font></span></a>, <a class="txt" href="#E47:162_1"><span class="lab"><font color="Green" title="E106">A34</font></span></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></span>;<br/>
<a NAME="E52:162_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:162"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E5:162_1"><span class="lab"><font color="Green" title="E81">A9</font></span></a>, <a class="txt" href="#E27:162_1"><span class="lab"><font color="Green" title="E94">A22</font></span></a>, <a class="txt" href="#E50:162_1"><span class="lab"><font color="Green" title="E109">A37</font></span></a>, <a class="txt" href="#E16:162_1"><span class="lab"><font color="Green" title="E89">A17</font></span></a>, <a class="txt" href="#E23:162_1"><span class="lab"><font color="Green" title="E92">A20</font></span></a>, <a class="txt" href="#E49:162_1"><span class="lab"><font color="Green" title="E108">A36</font></span></a>, <a class="txt" href="#E38:162_1"><span class="lab"><font color="Green" title="E102">A30</font></span></a>, <a class="txt" href="#E45:162_1"><span class="lab"><font color="Green" title="E105">A33</font></span></a>, <a class="txt" href="#E48:162_1"><span class="lab"><font color="Green" title="E107">A35</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E53:162_1"/>
<font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162_1"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="txt" href="#E7:162_1"><span class="lab"><font color="Green" title="E83">A11</font></span></a>, <a class="txt" href="#E28:162_1"><span class="lab"><font color="Green" title="E95">A23</font></span></a>, <a class="txt" href="#E29:162_1"><span class="lab"><font color="Green" title="E96">A24</font></span></a>, <a class="txt" href="#E22:162_1"><span class="lab"><font color="Green" title="E91">A19</font></span></a>, <a class="txt" href="#E12:162_1"><span class="lab"><font color="Green" title="E87">A15</font></span></a>, <a class="txt" href="#E33:162_1"><span class="lab"><font color="Green" title="E99">A27</font></span></a>, <a class="txt" href="#E44:162_1"><span class="lab"><font color="Green" title="E104">A32</font></span></a>, <a class="txt" href="#E34:162_1"><span class="lab"><font color="Green" title="E100">A28</font></span></a>, <a class="txt" href="#E48:162_1"><span class="lab"><font color="Green" title="E107">A35</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="162_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> ) &amp; ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:162_2_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:162_2_1"/><span class="lab"><font color="Green" title="E78">A38</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:162_2_1"/><span class="lab"><font color="Green" title="E79">A39</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c9">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162_2_1"><span class="lab"><font color="Green" title="E78">A38</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c12">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:162_2_1"/><span class="lab"><font color="Green" title="E80">A40</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E8:162_2_1"/><span class="lab"><font color="Green" title="E81">A41</font></span>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:162_2_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">j</font>
 <span class="kw">and </span><br/><a NAME="E10:162_2_1"/><span class="lab"><font color="Green" title="E82">A42</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162_2_1"><span class="lab"><font color="Green" title="E78">A38</font></span></a>, <a class="ref" href="pre_poly.html#T29" target="_self" title="PRE_POLY:th.29">Th29</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">dbi</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c11">i</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c14">ddbi1</font> =  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>;<br/><a NAME="E12:162_2_1"/><span class="lab"><font color="Green" title="E83">A43</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="txt" href="#E7:162_2_1"><span class="lab"><font color="Green" title="E80">A40</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A44</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>

;<br/><a NAME="E14:162_2_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E15:162_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A45</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:162_2_1"><span class="lab"><font color="Green" title="E81">A41</font></span></a>, <a class="txt" href="#E13:162_2_1"><span class="lab"><font color="Green" title="E84">A44</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c16">b2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E17:162_2_1"/><span class="lab"><font color="Green" title="E86">A46</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E18:162_2_1"/><span class="lab"><font color="Green" title="E87">A47</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E7:162_2_1"><span class="lab"><font color="Green" title="E80">A40</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">b19</font> = <font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c18">b29</font> = <font color="Maroon" title="c16">b2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" target="_self" title="PRE_POLY:def.12">Def12</a></span>;<br/><a NAME="E20:162_2_1"/><span class="lab"><font color="Green" title="E88">A48</font></span>: 
( <font color="Maroon" title="c17">b19</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font> &amp; <font color="Maroon" title="c18">b29</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">b2</font> )
 ;<br/><a NAME="E21:162_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A49</font></span>: 
<font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:162_2_1"><span class="lab"><font color="Green" title="E86">A46</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><a NAME="E22:162_2_1"/>
<font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:162_2_1"><span class="lab"><font color="Green" title="E86">A46</font></span></a>, <a class="txt" href="#E20:162_2_1"><span class="lab"><font color="Green" title="E88">A48</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">b11</font>, <font color="Maroon" title="c20">b12</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E24:162_2_1"/><span class="lab"><font color="Green" title="E90">A50</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c19">b11</font>,<font color="Maroon" title="c20">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E25:162_2_1"/><span class="lab"><font color="Green" title="E91">A51</font></span>: 
<font color="Maroon" title="c15">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">b11</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c20">b12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:162_2_1"><span class="lab"><font color="Green" title="E81">A41</font></span></a>, <a class="txt" href="#E13:162_2_1"><span class="lab"><font color="Green" title="E84">A44</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/><a NAME="E26:162_2_1"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">b11</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">b12</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:162_2_1"><span class="lab"><font color="Green" title="E87">A47</font></span></a>, <a class="txt" href="#E25:162_2_1"><span class="lab"><font color="Green" title="E91">A51</font></span></a>, <a class="ref" href="pre_poly.html#T35" target="_self" title="PRE_POLY:th.35">Th35</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">i9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E28:162_2_1"/><span class="lab"><font color="Green" title="E92">A52</font></span>: 
<font color="Maroon" title="c21">i9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E29:162_2_1"/><span class="lab"><font color="Green" title="E93">A53</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c21">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c19">b11</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">b12</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T69" target="_self" title="PRE_POLY:th.69">Th69</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">b3</font> = <font color="Maroon" title="c20">b12</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">b119</font> = <font color="Maroon" title="c19">b11</font>, <font color="Maroon" title="c24">b39</font> = <font color="Maroon" title="c20">b12</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" target="_self" title="PRE_POLY:def.12">Def12</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c25">j9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E32:162_2_1"/><span class="lab"><font color="Green" title="E94">A54</font></span>: 
<font color="Maroon" title="c25">j9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b12</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E33:162_2_1"/><span class="lab"><font color="Green" title="E95">A55</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b12</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font></span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c25">j9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c20">b12</font>,<font color="Maroon" title="c16">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T69" target="_self" title="PRE_POLY:th.69">Th69</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">dbi9</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c21">i9</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c27">ddbi92</font> =  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>;<br/><a NAME="E35:162_2_1"/><span class="lab"><font color="Green" title="E96">A56</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c21">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c23">b119</font>,<font color="Maroon" title="c24">b39</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:162_2_1"><span class="lab"><font color="Green" title="E93">A53</font></span></a></span>;<br/><a NAME="E36:162_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A57</font></span>: 
( <font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">b11</font> &amp;  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">b12</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><a NAME="E37:162_2_1"/><span class="lab"><font color="Green" title="E98">A58</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:162"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E4:162"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="txt" href="#E28:162_2_1"><span class="lab"><font color="Green" title="E92">A52</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A59</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i9</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>

;<br/><a NAME="E39:162_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A60</font></span>: 
<font color="Maroon" title="c25">j9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:162_2_1"><span class="lab"><font color="Green" title="E94">A54</font></span></a>, <a class="txt" href="#E35:162_2_1"><span class="lab"><font color="Green" title="E96">A56</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><a NAME="E40:162_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A61</font></span>: 
<font color="Maroon" title="c25">j9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:162_2_1"><span class="lab"><font color="Green" title="E99">A59</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E41:162_2_1"/><span class="lab"><font color="Green" title="E102">A62</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b12</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font></span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c25">j9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b12</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:162_2_1"><span class="lab"><font color="Green" title="E94">A54</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c28">m</font> = <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i9</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">j9</font>;<br/><a NAME="E42:162_2_1"/><span class="lab"><font color="Green" title="E103">A63</font></span>: 
( <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i9</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">j9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</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="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c21">i9</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">j9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:162"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E28:162_2_1"><span class="lab"><font color="Green" title="E92">A52</font></span></a>, <a class="txt" href="#E39:162_2_1"><span class="lab"><font color="Green" title="E100">A60</font></span></a>, <a class="ref" href="pre_poly.html#T30" target="_self" title="PRE_POLY:th.30">Th30</a></span>;<br/><span class="lab"><font color="Green" title="E104">A64</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j9</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:162_2_1"><span class="lab"><font color="Green" title="E98">A58</font></span></a>, <a class="txt" href="#E39:162_2_1"><span class="lab"><font color="Green" title="E100">A60</font></span></a>, <a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c19">b11</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c20">b12</font>,<font color="Maroon" title="c16">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:162_2_1"><span class="lab"><font color="Green" title="E95">A55</font></span></a>, <a class="txt" href="#E36:162_2_1"><span class="lab"><font color="Green" title="E97">A57</font></span></a>, <a class="txt" href="#E40:162_2_1"><span class="lab"><font color="Green" title="E101">A61</font></span></a>, <a class="txt" href="#E41:162_2_1"><span class="lab"><font color="Green" title="E102">A62</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c19">b11</font>,<font color="Maroon" title="c20">b12</font>,<font color="Maroon" title="c16">b2</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>
;<br/><a NAME="E44:162_2_1"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:162_2_1"><span class="lab"><font color="Green" title="E81">A41</font></span></a>, <a class="txt" href="#E13:162_2_1"><span class="lab"><font color="Green" title="E84">A44</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c19">b11</font>,<font color="Maroon" title="c20">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c16">b2</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:162_2_1"><span class="lab"><font color="Green" title="E81">A41</font></span></a>, <a class="txt" href="#E12:162_2_1"><span class="lab"><font color="Green" title="E83">A43</font></span></a>, <a class="txt" href="#E21:162_2_1"><span class="lab"><font color="Green" title="E89">A49</font></span></a>, <a class="txt" href="#E24:162_2_1"><span class="lab"><font color="Green" title="E90">A50</font></span></a>, <a class="txt" href="#E15:162_2_1"><span class="lab"><font color="Green" title="E85">A45</font></span></a>, <a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c19">b11</font>,<font color="Maroon" title="c20">b12</font>,<font color="Maroon" title="c16">b2</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>
;<br/><span class="kw">hence </span><a NAME="E46:162_2_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:162_2_1"><span class="lab"><font color="Green" title="E79">A39</font></span></a>, <a class="txt" href="#E10:162_2_1"><span class="lab"><font color="Green" title="E82">A42</font></span></a>, <a class="txt" href="#E43:162_2_1"><span class="lab"><font color="Green" title="E104">A64</font></span></a>, <a class="txt" href="#E42:162_2_1"><span class="lab"><font color="Green" title="E103">A63</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:162_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:162_2"/><span class="lab"><font color="Green" title="E78">A65</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:162_2"/><span class="lab"><font color="Green" title="E79">A66</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c9">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:162_2"><span class="lab"><font color="Green" title="E78">A65</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c12">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:162_2"/><span class="lab"><font color="Green" title="E80">A67</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">and </span><br/><a NAME="E9:162_2"/><span class="lab"><font color="Green" title="E81">A68</font></span>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:162_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">j</font>
 <span class="kw">and </span><br/><a NAME="E11:162_2"/><span class="lab"><font color="Green" title="E82">A69</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:162_2"><span class="lab"><font color="Green" title="E78">A65</font></span></a>, <a class="ref" href="pre_poly.html#T29" target="_self" title="PRE_POLY:th.29">Th29</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">dbi</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c11">i</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c14">ddbi1</font> =  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>;<br/><a NAME="E13:162_2"/><span class="lab"><font color="Green" title="E83">A70</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:162"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="txt" href="#E8:162_2"><span class="lab"><font color="Green" title="E80">A67</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A71</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>

;<br/><a NAME="E15:162_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E16:162_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A72</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:162_2"><span class="lab"><font color="Green" title="E81">A68</font></span></a>, <a class="txt" href="#E14:162_2"><span class="lab"><font color="Green" title="E84">A71</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c16">b2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E18:162_2"/><span class="lab"><font color="Green" title="E86">A73</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E19:162_2"/><span class="lab"><font color="Green" title="E87">A74</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c16">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:162"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E8:162_2"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">b19</font> = <font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c18">b29</font> = <font color="Maroon" title="c16">b2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" target="_self" title="PRE_POLY:def.12">Def12</a></span>;<br/><a NAME="E21:162_2"/><span class="lab"><font color="Green" title="E88">A75</font></span>: 
( <font color="Maroon" title="c17">b19</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font> &amp; <font color="Maroon" title="c18">b29</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">b2</font> )
 ;<br/><a NAME="E22:162_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A76</font></span>: 
<font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:162_2"><span class="lab"><font color="Green" title="E86">A73</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><a NAME="E23:162_2"/>
<font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:162_2"><span class="lab"><font color="Green" title="E86">A73</font></span></a>, <a class="txt" href="#E21:162_2"><span class="lab"><font color="Green" title="E88">A75</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">b11</font>, <font color="Maroon" title="c20">b12</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E25:162_2"/><span class="lab"><font color="Green" title="E90">A77</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c19">b11</font>,<font color="Maroon" title="c20">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E26:162_2"/><span class="lab"><font color="Green" title="E91">A78</font></span>: 
<font color="Maroon" title="c16">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">b11</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c20">b12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:162_2"><span class="lab"><font color="Green" title="E81">A68</font></span></a>, <a class="txt" href="#E14:162_2"><span class="lab"><font color="Green" title="E84">A71</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/><a NAME="E27:162_2"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">dbi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:162_2"><span class="lab"><font color="Green" title="E81">A68</font></span></a>, <a class="txt" href="#E14:162_2"><span class="lab"><font color="Green" title="E84">A71</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A79</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> = 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c15">b1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c19">b11</font>,<font color="Maroon" title="c20">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:162_2"><span class="lab"><font color="Green" title="E81">A68</font></span></a>, <a class="txt" href="#E13:162_2"><span class="lab"><font color="Green" title="E83">A70</font></span></a>, <a class="txt" href="#E22:162_2"><span class="lab"><font color="Green" title="E89">A76</font></span></a>, <a class="txt" href="#E25:162_2"><span class="lab"><font color="Green" title="E90">A77</font></span></a>, <a class="txt" href="#E16:162_2"><span class="lab"><font color="Green" title="E85">A72</font></span></a>, <a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c19">b11</font>,<font color="Maroon" title="c20">b12</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>
;<br/><span class="kw">set </span><font color="Maroon" title="c21">b3</font> = <font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b11</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">b39</font> = <font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b11</font>, <font color="Maroon" title="c23">b129</font> = <font color="Maroon" title="c20">b12</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" target="_self" title="PRE_POLY:def.12">Def12</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">j9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E31:162_2"/><span class="lab"><font color="Green" title="E93">A80</font></span>: 
<font color="Maroon" title="c24">j9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b11</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E32:162_2"/><span class="lab"><font color="Green" title="E94">A81</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b11</font></span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c24">j9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c19">b11</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T69" target="_self" title="PRE_POLY:th.69">Th69</a></span>;<br/><a NAME="E33:162_2"/><span class="lab"><font color="Green" title="E95">A82</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b11</font></span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c24">j9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b11</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:162_2"><span class="lab"><font color="Green" title="E93">A80</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E34:162_2"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b11</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c20">b12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:162_2"><span class="lab"><font color="Green" title="E87">A74</font></span></a>, <a class="txt" href="#E26:162_2"><span class="lab"><font color="Green" title="E91">A78</font></span></a>, <a class="ref" href="pre_poly.html#T35" target="_self" title="PRE_POLY:th.35">Th35</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">i9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E36:162_2"/><span class="lab"><font color="Green" title="E96">A83</font></span>: 
<font color="Maroon" title="c25">i9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E37:162_2"/><span class="lab"><font color="Green" title="E97">A84</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c25">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b11</font></span>)</span>,<font color="Maroon" title="c20">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T69" target="_self" title="PRE_POLY:th.69">Th69</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">dbi9</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c25">i9</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c27">ddbi92</font> =  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c28">m</font> = <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">i9</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">j9</font>;<br/><a NAME="E39:162_2"/><span class="lab"><font color="Green" title="E98">A85</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c25">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c22">b39</font>,<font color="Maroon" title="c23">b129</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:162_2"><span class="lab"><font color="Green" title="E97">A84</font></span></a></span>;<br/><a NAME="E40:162_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A86</font></span>: 
<font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b11</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><a NAME="E41:162_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A87</font></span>: 
<font color="Maroon" title="c24">j9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:162_2"><span class="lab"><font color="Green" title="E93">A80</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E42:162_2"/><span class="lab"><font color="Green" title="E101">A88</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E3:162"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="txt" href="#E36:162_2"><span class="lab"><font color="Green" title="E96">A83</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A89</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i9</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>

;<br/><a NAME="E44:162_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A90</font></span>: 
( <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">i9</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">j9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</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="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c25">i9</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">j9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E36:162_2"><span class="lab"><font color="Green" title="E96">A83</font></span></a>, <a class="txt" href="#E31:162_2"><span class="lab"><font color="Green" title="E93">A80</font></span></a>, <a class="txt" href="#E40:162_2"><span class="lab"><font color="Green" title="E99">A86</font></span></a>, <a class="ref" href="pre_poly.html#T30" target="_self" title="PRE_POLY:th.30">Th30</a></span>;<br/><a NAME="E45:162_2"/><span class="lab"><font color="Green" title="E104">A91</font></span>: 
<font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c20">b12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:162_2"><span class="lab"><font color="Green" title="E98">A85</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j9</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">dbi9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:162_2"><span class="lab"><font color="Green" title="E93">A80</font></span></a>, <a class="txt" href="#E42:162_2"><span class="lab"><font color="Green" title="E101">A88</font></span></a>, <a class="txt" href="#E40:162_2"><span class="lab"><font color="Green" title="E99">A86</font></span></a>, <a class="txt" href="#E43:162_2"><span class="lab"><font color="Green" title="E102">A89</font></span></a>, <a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c19">b11</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c20">b12</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:162_2"><span class="lab"><font color="Green" title="E94">A81</font></span></a>, <a class="txt" href="#E45:162_2"><span class="lab"><font color="Green" title="E104">A91</font></span></a>, <a class="txt" href="#E40:162_2"><span class="lab"><font color="Green" title="E99">A86</font></span></a>, <a class="txt" href="#E41:162_2"><span class="lab"><font color="Green" title="E100">A87</font></span></a>, <a class="txt" href="#E33:162_2"><span class="lab"><font color="Green" title="E95">A82</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c19">b11</font>,<font color="Maroon" title="c20">b12</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>
;<br/><span class="kw">hence </span><a NAME="E47:162_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:162_2"><span class="lab"><font color="Green" title="E79">A66</font></span></a>, <a class="txt" href="#E11:162_2"><span class="lab"><font color="Green" title="E82">A69</font></span></a>, <a class="txt" href="#E28:162_2"><span class="lab"><font color="Green" title="E92">A79</font></span></a>, <a class="txt" href="#E44:162_2"><span class="lab"><font color="Green" title="E103">A90</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:162"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A92</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<a NAME="E8:162"/>
 <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="162_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k1</font>, <font color="Maroon" title="c9">k2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <font color="Maroon" title="c9">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k2</font> or <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:162_3"/><span class="lab"><font color="Green" title="E79">A93</font></span>: 
<font color="Maroon" title="c8">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:162_3"/><span class="lab"><font color="Green" title="E80">A94</font></span>: 
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:162_3"/><span class="lab"><font color="Green" title="E81">A95</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">i1</font>, <font color="Maroon" title="c11">j1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:162_3"/><span class="lab"><font color="Green" title="E82">A96</font></span>: 
<font color="Maroon" title="c10">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E6:162_3"/><span class="lab"><font color="Green" title="E83">A97</font></span>: 
<font color="Maroon" title="c11">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:162_3"/><span class="lab"><font color="Green" title="E84">A98</font></span>: 
<font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">j1</font>
 <span class="kw">and </span><br/><a NAME="E8:162_3"/><span class="lab"><font color="Green" title="E85">A99</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162_3"><span class="lab"><font color="Green" title="E79">A93</font></span></a>, <a class="ref" href="pre_poly.html#T29" target="_self" title="PRE_POLY:th.29">Th29</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">dbi1</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c10">i1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">ddbi11</font> =  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>;<br/>
<a NAME="E10:162_3"/><span class="lab"><font color="Green" title="E86">A100</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="txt" href="#E5:162_3"><span class="lab"><font color="Green" title="E82">A96</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E87">A101</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>

;<br/>
<a NAME="E12:162_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A102</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162_3"><span class="lab"><font color="Green" title="E83">A97</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E13:162_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E14:162_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A103</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162_3"><span class="lab"><font color="Green" title="E83">A97</font></span></a>, <a class="txt" href="#E11:162_3"><span class="lab"><font color="Green" title="E87">A101</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">b11</font>, <font color="Maroon" title="c15">b12</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E16:162_3"/><span class="lab"><font color="Green" title="E90">A104</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c14">b11</font>,<font color="Maroon" title="c15">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E17:162_3"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b11</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c15">b12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E5:162_3"><span class="lab"><font color="Green" title="E82">A96</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">b119</font> = <font color="Maroon" title="c14">b11</font>, <font color="Maroon" title="c17">b129</font> = <font color="Maroon" title="c15">b12</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" target="_self" title="PRE_POLY:def.12">Def12</a></span>;<br/>
<a NAME="E19:162_3"/><span class="lab"><font color="Green" title="E91">A105</font></span>: 
( <font color="Maroon" title="c16">b119</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b11</font> &amp; <font color="Maroon" title="c17">b129</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b12</font> )
 
;<br/>
<a NAME="E20:162_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:162_3"><span class="lab"><font color="Green" title="E90">A104</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">b111</font>, <font color="Maroon" title="c19">b112</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E22:162_3"/><span class="lab"><font color="Green" title="E92">A106</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c18">b111</font>,<font color="Maroon" title="c19">b112</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E23:162_3"/><span class="lab"><font color="Green" title="E93">A107</font></span>: 
<font color="Maroon" title="c14">b11</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b111</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c19">b112</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162_3"><span class="lab"><font color="Green" title="E83">A97</font></span></a>, <a class="txt" href="#E11:162_3"><span class="lab"><font color="Green" title="E87">A101</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/>
<a NAME="E24:162_3"/>
<font color="Maroon" title="c12">dbi1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:162_3"><span class="lab"><font color="Green" title="E90">A104</font></span></a>, <a class="txt" href="#E19:162_3"><span class="lab"><font color="Green" title="E91">A105</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E94">A108</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j1</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c18">b111</font>,<font color="Maroon" title="c19">b112</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c15">b12</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162_3"><span class="lab"><font color="Green" title="E83">A97</font></span></a>, <a class="txt" href="#E10:162_3"><span class="lab"><font color="Green" title="E86">A100</font></span></a>, <a class="txt" href="#E22:162_3"><span class="lab"><font color="Green" title="E92">A106</font></span></a>, <a class="txt" href="#E12:162_3"><span class="lab"><font color="Green" title="E88">A102</font></span></a>, <a class="txt" href="#E14:162_3"><span class="lab"><font color="Green" title="E89">A103</font></span></a>, <a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c18">b111</font>,<font color="Maroon" title="c19">b112</font>,<font color="Maroon" title="c15">b12</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">i2</font>, <font color="Maroon" title="c21">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E27:162_3"/><span class="lab"><font color="Green" title="E95">A109</font></span>: 
<font color="Maroon" title="c20">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E28:162_3"/><span class="lab"><font color="Green" title="E96">A110</font></span>: 
<font color="Maroon" title="c21">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E29:162_3"/><span class="lab"><font color="Green" title="E97">A111</font></span>: 
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K10" title="PRE_POLY:func.10">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">j2</font>
 <span class="kw">and </span><br/><a NAME="E30:162_3"/><span class="lab"><font color="Green" title="E98">A112</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:162_3"><span class="lab"><font color="Green" title="E80">A94</font></span></a>, <a class="ref" href="pre_poly.html#T29" target="_self" title="PRE_POLY:th.29">Th29</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">dbi2</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c20">i2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">ddbi21</font> =  <a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>;<br/>
<a NAME="E32:162_3"/><span class="lab"><font color="Green" title="E99">A113</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K8" title="PRE_POLY:func.8">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="txt" href="#E27:162_3"><span class="lab"><font color="Green" title="E95">A109</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E100">A114</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p5">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>

;<br/>
<a NAME="E34:162_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A115</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:162_3"><span class="lab"><font color="Green" title="E96">A110</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E35:162_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E36:162_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A116</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:162_3"><span class="lab"><font color="Green" title="E96">A110</font></span></a>, <a class="txt" href="#E33:162_3"><span class="lab"><font color="Green" title="E100">A114</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">b21</font>, <font color="Maroon" title="c25">b22</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E38:162_3"/><span class="lab"><font color="Green" title="E103">A117</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c24">b21</font>,<font color="Maroon" title="c25">b22</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E39:162_3"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b21</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c25">b22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E27:162_3"><span class="lab"><font color="Green" title="E95">A109</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">b219</font> = <font color="Maroon" title="c24">b21</font>, <font color="Maroon" title="c27">b229</font> = <font color="Maroon" title="c25">b22</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" target="_self" title="PRE_POLY:def.12">Def12</a></span>;<br/>
<a NAME="E41:162_3"/><span class="lab"><font color="Green" title="E104">A118</font></span>: 
( <font color="Maroon" title="c26">b219</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b21</font> &amp; <font color="Maroon" title="c27">b229</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">b22</font> )
 
;<br/>
<a NAME="E42:162_3"/><span class="kw">then </span>
<font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b21</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:162_3"><span class="lab"><font color="Green" title="E103">A117</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">b211</font>, <font color="Maroon" title="c29">b212</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E44:162_3"/><span class="lab"><font color="Green" title="E105">A119</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c21">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c28">b211</font>,<font color="Maroon" title="c29">b212</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E45:162_3"/><span class="lab"><font color="Green" title="E106">A120</font></span>: 
<font color="Maroon" title="c24">b21</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c28">b211</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c29">b212</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:162_3"><span class="lab"><font color="Green" title="E96">A110</font></span></a>, <a class="txt" href="#E33:162_3"><span class="lab"><font color="Green" title="E100">A114</font></span></a>, <a class="ref" href="pre_poly.html#T68" target="_self" title="PRE_POLY:th.68">Th68</a></span>;<br/>
<a NAME="E46:162_3"/>
<font color="Maroon" title="c22">dbi2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">b22</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:162_3"><span class="lab"><font color="Green" title="E103">A117</font></span></a>, <a class="txt" href="#E41:162_3"><span class="lab"><font color="Green" title="E104">A118</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E107">A121</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j2</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c28">b211</font>,<font color="Maroon" title="c29">b212</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c25">b22</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:162_3"><span class="lab"><font color="Green" title="E96">A110</font></span></a>, <a class="txt" href="#E32:162_3"><span class="lab"><font color="Green" title="E99">A113</font></span></a>, <a class="txt" href="#E44:162_3"><span class="lab"><font color="Green" title="E105">A119</font></span></a>, <a class="txt" href="#E34:162_3"><span class="lab"><font color="Green" title="E101">A115</font></span></a>, <a class="txt" href="#E36:162_3"><span class="lab"><font color="Green" title="E102">A116</font></span></a>, <a class="ref" href="pre_poly.html#D4" target="_self" title="PRE_POLY:def.4">Def4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c28">b211</font>,<font color="Maroon" title="c29">b212</font>,<font color="Maroon" title="c25">b22</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>
;<br/>
<a NAME="E48:162_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A122</font></span>: 
( <font color="Maroon" title="c18">b111</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c28">b211</font> &amp; <font color="Maroon" title="c19">b112</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c29">b212</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162_3"><span class="lab"><font color="Green" title="E81">A95</font></span></a>, <a class="txt" href="#E8:162_3"><span class="lab"><font color="Green" title="E85">A99</font></span></a>, <a class="txt" href="#E30:162_3"><span class="lab"><font color="Green" title="E98">A112</font></span></a>, <a class="txt" href="#E25:162_3"><span class="lab"><font color="Green" title="E94">A108</font></span></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></span>;<br/>
<a NAME="E49:162_3"/><span class="lab"><font color="Green" title="E109">A123</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c20">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E27:162_3"><span class="lab"><font color="Green" title="E95">A109</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E50:162_3"/><span class="lab"><font color="Green" title="E110">A124</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E5:162_3"><span class="lab"><font color="Green" title="E82">A96</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E51:162_3"/>
<font color="Maroon" title="c15">b12</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">b22</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:162_3"><span class="lab"><font color="Green" title="E81">A95</font></span></a>, <a class="txt" href="#E8:162_3"><span class="lab"><font color="Green" title="E85">A99</font></span></a>, <a class="txt" href="#E30:162_3"><span class="lab"><font color="Green" title="E98">A112</font></span></a>, <a class="txt" href="#E25:162_3"><span class="lab"><font color="Green" title="E94">A108</font></span></a>, <a class="txt" href="#E47:162_3"><span class="lab"><font color="Green" title="E107">A121</font></span></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></span>;<br/>
<a NAME="E52:162_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:162"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E5:162_3"><span class="lab"><font color="Green" title="E82">A96</font></span></a>, <a class="txt" href="#E27:162_3"><span class="lab"><font color="Green" title="E95">A109</font></span></a>, <a class="txt" href="#E50:162_3"><span class="lab"><font color="Green" title="E110">A124</font></span></a>, <a class="txt" href="#E16:162_3"><span class="lab"><font color="Green" title="E90">A104</font></span></a>, <a class="txt" href="#E23:162_3"><span class="lab"><font color="Green" title="E93">A107</font></span></a>, <a class="txt" href="#E49:162_3"><span class="lab"><font color="Green" title="E109">A123</font></span></a>, <a class="txt" href="#E38:162_3"><span class="lab"><font color="Green" title="E103">A117</font></span></a>, <a class="txt" href="#E45:162_3"><span class="lab"><font color="Green" title="E106">A120</font></span></a>, <a class="txt" href="#E48:162_3"><span class="lab"><font color="Green" title="E108">A122</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E53:162_3"/>
<font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:162_3"><span class="lab"><font color="Green" title="E83">A97</font></span></a>, <a class="txt" href="#E7:162_3"><span class="lab"><font color="Green" title="E84">A98</font></span></a>, <a class="txt" href="#E28:162_3"><span class="lab"><font color="Green" title="E96">A110</font></span></a>, <a class="txt" href="#E29:162_3"><span class="lab"><font color="Green" title="E97">A111</font></span></a>, <a class="txt" href="#E22:162_3"><span class="lab"><font color="Green" title="E92">A106</font></span></a>, <a class="txt" href="#E12:162_3"><span class="lab"><font color="Green" title="E88">A102</font></span></a>, <a class="txt" href="#E33:162_3"><span class="lab"><font color="Green" title="E100">A114</font></span></a>, <a class="txt" href="#E44:162_3"><span class="lab"><font color="Green" title="E105">A119</font></span></a>, <a class="txt" href="#E34:162_3"><span class="lab"><font color="Green" title="E101">A115</font></span></a>, <a class="txt" href="#E48:162_3"><span class="lab"><font color="Green" title="E108">A122</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:162"/><span class="kw">then </span>
 <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font>, <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:162"><span class="lab"><font color="Green" title="E78">A92</font></span></a>, <a class="txt" href="#E5:162"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="ref" href="rfinseq.html#T26" title="RFINSEQ:th.26">RFINSEQ:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:162"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> st  <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T4" title="RFINSEQ:th.4">RFINSEQ:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
