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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">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="polynom1.html#K12" title="POLYNOM1:func.12">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="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="polynom1.html#K3" title="POLYNOM1:func.3">^^</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="binarith.html#K10" title="BINARITH:func.10">*&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b2">f</font></span>)</span> st  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="dtconstr.html#K15" title="DTCONSTR:func.15">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/><b>let </b><font color="Maroon" title="c2">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="polynom1.html#K12" title="POLYNOM1:func.12">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="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="polynom1.html#K3" title="POLYNOM1:func.3">^^</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="binarith.html#K10" title="BINARITH:func.10">*&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b1">f</font></span>)</span> st  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="dtconstr.html#K15" title="DTCONSTR:func.15">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/><b>let </b><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="polynom1.html#K12" title="POLYNOM1:func.12">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:()"><i><font color="Red">::  thesis: </font></i></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="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="polynom1.html#K3" title="POLYNOM1:func.3">^^</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="binarith.html#K10" title="BINARITH:func.10">*&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> st  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="dtconstr.html#K15" title="DTCONSTR:func.15">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/>





<b>assume </b><b>that </b><br/><a NAME="E1:142"/><i><font color="Green" title="E85">A1</font></i>: 
 <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>and </b><br/><a NAME="E2:142"/><i><font color="Green" title="E86">A2</font></i>: 
 <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>and </b><br/><a NAME="E3:142"/><i><font color="Green" title="E87">A3</font></i>: 
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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="polynom1.html#K3" title="POLYNOM1:func.3">^^</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:142"/><i><font color="Green" title="E88">A4</font></i>: 
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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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:()"><i><font color="Red">::  thesis: </font></i></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  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> st  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="dtconstr.html#K15" title="DTCONSTR:func.15">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/>

<b>set </b><font color="Maroon" title="c5">Ff</font> =  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Fg</font> =  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font>;<br/>
<b>set </b><font color="Maroon" title="c7">db</font> =  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> )</span>
<div class="add"><b>assume </b><a NAME="E1:142_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c9">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:142_1_1"/><i><font color="Green" title="E89">A5</font></i>: 
( <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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <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="dtconstr.html#K15" title="DTCONSTR:func.15">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> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><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>  <b>by </b><i><a class="txt" href="#E3:142_1_1"><i><font color="Green" title="E89">A5</font></i></a></i>;<br/><b>consider </b><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> <b> such that </b><br/><a NAME="E6:142_1_1"/><i><font color="Green" title="E90">A6</font></i>: 
<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>
 <b>and </b><br/><a NAME="E7:142_1_1"/><i><font color="Green" title="E91">A7</font></i>: 
<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>
 <b>and </b><br/><a NAME="E8:142_1_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="polynom1.html#K4" title="POLYNOM1:func.4">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>
 <b>and </b><br/><a NAME="E9:142_1_1"/><i><font color="Green" title="E92">A8</font></i>: 
<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="dtconstr.html#K15" title="DTCONSTR:func.15">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>
 <b>by </b><i><a class="txt" href="#E3:142_1_1"><i><font color="Green" title="E89">A5</font></i></a>, <a class="ref" href="polynom1.html#T32" target="_self" title="POLYNOM1:th.32">Th32</a></i>;<br/><b>set </b><font color="Maroon" title="c13">ddbi1</font> =  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>;<br/><a NAME="E10:142_1_1"/><i><font color="Green" title="E93">A9</font></i>: 
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:142"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="#E6:142_1_1"><i><font color="Green" title="E90">A6</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c14">b1</font>, <font color="Maroon" title="c15">b2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E12:142_1_1"/><i><font color="Green" title="E94">A10</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="c14">b1</font>,<font color="Maroon" title="c15">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E13:142_1_1"/><i><font color="Green" title="E95">A11</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font>
 <b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E6:142_1_1"><i><font color="Green" title="E90">A6</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">b1'</font> = <font color="Maroon" title="c14">b1</font>, <font color="Maroon" title="c17">b2'</font> = <font color="Maroon" title="c15">b2</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/><a NAME="E15:142_1_1"/><i><font color="Green" title="E96">A12</font></i>: 
( <font color="Maroon" title="c16">b1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b1</font> &amp; <font color="Maroon" title="c17">b2'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b2</font> )
 ;<br/><a NAME="E16:142_1_1"/><b>then </b><i><font color="Green" title="E97">A13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b1</font>
 <b>by </b><i><a class="txt" href="#E12:142_1_1"><i><font color="Green" title="E94">A10</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><a NAME="E17:142_1_1"/><i><font color="Green" title="E98">A14</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b2</font>
 <b>by </b><i><a class="txt" href="#E12:142_1_1"><i><font color="Green" title="E94">A10</font></i></a>, <a class="txt" href="#E15:142_1_1"><i><font color="Green" title="E96">A12</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><i><font color="Green" title="E99">A15</font></i>:  <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E10:142_1_1"><i><font color="Green" title="E93">A9</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>

;<br/><b>then consider </b><font color="Maroon" title="c18">b11</font>, <font color="Maroon" title="c19">b12</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E20:142_1_1"/><i><font color="Green" title="E100">A16</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="c18">b11</font>,<font color="Maroon" title="c19">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E21:142_1_1"/><i><font color="Green" title="E101">A17</font></i>: 
<font color="Maroon" title="c14">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b11</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c19">b12</font>
 <b>by </b><i><a class="txt" href="#E7:142_1_1"><i><font color="Green" title="E91">A7</font></i></a>, <a class="txt" href="#E16:142_1_1"><i><font color="Green" title="E97">A13</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/><a NAME="E22:142_1_1"/><i><font color="Green" title="E102">A18</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E23:142_1_1"/><i><font color="Green" title="E103">A19</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <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>
 <b>by </b><i><a class="txt" href="#E7:142_1_1"><i><font color="Green" title="E91">A7</font></i></a>, <a class="txt" href="#E18:142_1_1"><i><font color="Green" title="E99">A15</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E24:142_1_1"/>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&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="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E7:142_1_1"><i><font color="Green" title="E91">A7</font></i></a>, <a class="txt" href="#E18:142_1_1"><i><font color="Green" title="E99">A15</font></i></a>, <a class="txt" href="#E22:142_1_1"><i><font color="Green" title="E102">A18</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>then </b><i><font color="Green" title="E104">A20</font></i>: <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="c18">b11</font>,<font color="Maroon" title="c19">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="c15">b2</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E7:142_1_1"><i><font color="Green" title="E91">A7</font></i></a>, <a class="txt" href="#E10:142_1_1"><i><font color="Green" title="E93">A9</font></i></a>, <a class="txt" href="#E17:142_1_1"><i><font color="Green" title="E98">A14</font></i></a>, <a class="txt" href="#E20:142_1_1"><i><font color="Green" title="E100">A16</font></i></a>, <a class="txt" href="#E23:142_1_1"><i><font color="Green" title="E103">A19</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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">b11</font>,<font color="Maroon" title="c19">b12</font>,<font color="Maroon" title="c15">b2</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>
;<br/><a NAME="E26:142_1_1"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b11</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">b12</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:142_1_1"><i><font color="Green" title="E95">A11</font></i></a>, <a class="txt" href="#E21:142_1_1"><i><font color="Green" title="E101">A17</font></i></a>, <a class="ref" href="polynom1.html#T39" target="_self" title="POLYNOM1:th.39">Th39</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">i'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E28:142_1_1"/><i><font color="Green" title="E105">A21</font></i>: 
<font color="Maroon" title="c20">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> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>and </b><br/><a NAME="E29:142_1_1"/><i><font color="Green" title="E106">A22</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">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="c18">b11</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">b12</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#T73" target="_self" title="POLYNOM1:th.73">Th73</a></i>;<br/><b>set </b><font color="Maroon" title="c21">b3</font> = <font color="Maroon" title="c19">b12</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font>;<br/><b>consider </b><font color="Maroon" title="c22">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> <b> such that </b><br/><a NAME="E31:142_1_1"/><i><font color="Green" title="E107">A23</font></i>: 
<font color="Maroon" title="c22">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"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">b12</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E32:142_1_1"/><i><font color="Green" title="E108">A24</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">b12</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font></span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c22">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">b12</font>,<font color="Maroon" title="c15">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#T73" target="_self" title="POLYNOM1:th.73">Th73</a></i>;<br/><b>set </b><font color="Maroon" title="c23">ddbi'2</font> =  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>;<br/><a NAME="E33:142_1_1"/><i><font color="Green" title="E109">A25</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="#E4:142"><i><font color="Green" title="E88">A4</font></i></a>, <a class="txt" href="#E28:142_1_1"><i><font color="Green" title="E105">A21</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">b11'</font> = <font color="Maroon" title="c18">b11</font>, <font color="Maroon" title="c25">b3'</font> = <font color="Maroon" title="c19">b12</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/><a NAME="E35:142_1_1"/><i><font color="Green" title="E110">A26</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c24">b11'</font>,<font color="Maroon" title="c25">b3'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E29:142_1_1"><i><font color="Green" title="E106">A22</font></i></a></i>;<br/><a NAME="E36:142_1_1"/><b>then </b><i><font color="Green" title="E111">A27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b11</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><a NAME="E37:142_1_1"/><i><font color="Green" title="E112">A28</font></i>: 
 <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">b12</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E35:142_1_1"><i><font color="Green" title="E110">A26</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><i><font color="Green" title="E113">A29</font></i>:  <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">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"><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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E33:142_1_1"><i><font color="Green" title="E109">A25</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>

;<br/><a NAME="E39:142_1_1"/><b>then </b><i><font color="Green" title="E114">A30</font></i>: 
<font color="Maroon" title="c22">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="c20">i'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E31:142_1_1"><i><font color="Green" title="E107">A23</font></i></a>, <a class="txt" href="#E35:142_1_1"><i><font color="Green" title="E110">A26</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><a NAME="E40:142_1_1"/><b>then </b><i><font color="Green" title="E115">A31</font></i>: 
<font color="Maroon" title="c22">j'</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E38:142_1_1"><i><font color="Green" title="E113">A29</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E41:142_1_1"/><i><font color="Green" title="E116">A32</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">b12</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font></span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c22">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">b12</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j'</font>
 <b>by </b><i><a class="txt" href="#E31:142_1_1"><i><font color="Green" title="E107">A23</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><i><font color="Green" title="E117">A33</font></i>: <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">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j'</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j'</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <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="c22">j'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E33:142_1_1"><i><font color="Green" title="E109">A25</font></i></a>, <a class="txt" href="#E39:142_1_1"><i><font color="Green" title="E114">A30</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c18">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="c19">b12</font>,<font color="Maroon" title="c15">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E32:142_1_1"><i><font color="Green" title="E108">A24</font></i></a>, <a class="txt" href="#E36:142_1_1"><i><font color="Green" title="E111">A27</font></i></a>, <a class="txt" href="#E37:142_1_1"><i><font color="Green" title="E112">A28</font></i></a>, <a class="txt" href="#E40:142_1_1"><i><font color="Green" title="E115">A31</font></i></a>, <a class="txt" href="#E41:142_1_1"><i><font color="Green" title="E116">A32</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<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">b11</font>,<font color="Maroon" title="c19">b12</font>,<font color="Maroon" title="c15">b2</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>
;<br/><b>set </b><font color="Maroon" title="c26">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="polynom1.html#K4" title="POLYNOM1:func.4">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">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="c22">j'</font>;<br/><a NAME="E43:142_1_1"/><i><font color="Green" title="E118">A34</font></i>: 
<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="polynom1.html#K4" title="POLYNOM1:func.4">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">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="c22">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"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="#E28:142_1_1"><i><font color="Green" title="E105">A21</font></i></a>, <a class="txt" href="#E39:142_1_1"><i><font color="Green" title="E114">A30</font></i></a>, <a class="ref" href="polynom1.html#T33" target="_self" title="POLYNOM1:th.33">Th33</a></i>;<br/><a NAME="E44:142_1_1"/>
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="polynom1.html#K4" title="POLYNOM1:func.4">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="c20">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="c22">j'</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="c20">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j'</font>
 <b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="#E28:142_1_1"><i><font color="Green" title="E105">A21</font></i></a>, <a class="txt" href="#E39:142_1_1"><i><font color="Green" title="E114">A30</font></i></a>, <a class="ref" href="polynom1.html#T33" target="_self" title="POLYNOM1:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E45:142_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:142_1_1"><i><font color="Green" title="E89">A5</font></i></a>, <a class="txt" href="#E9:142_1_1"><i><font color="Green" title="E92">A8</font></i></a>, <a class="txt" href="#E25:142_1_1"><i><font color="Green" title="E104">A20</font></i></a>, <a class="txt" href="#E42:142_1_1"><i><font color="Green" title="E117">A33</font></i></a>, <a class="txt" href="#E43:142_1_1"><i><font color="Green" title="E118">A34</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:142_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c9">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:142_1"/><i><font color="Green" title="E89">A35</font></i>: 
( <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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> &amp; <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="dtconstr.html#K15" title="DTCONSTR:func.15">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> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><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>  <b>by </b><i><a class="txt" href="#E4:142_1"><i><font color="Green" title="E89">A35</font></i></a></i>;<br/><b>consider </b><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> <b> such that </b><br/><a NAME="E7:142_1"/><i><font color="Green" title="E90">A36</font></i>: 
<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>
 <b>and </b><br/><a NAME="E8:142_1"/><i><font color="Green" title="E91">A37</font></i>: 
<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>
 <b>and </b><br/><a NAME="E9:142_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="polynom1.html#K4" title="POLYNOM1:func.4">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>
 <b>and </b><br/><a NAME="E10:142_1"/><i><font color="Green" title="E92">A38</font></i>: 
<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="dtconstr.html#K15" title="DTCONSTR:func.15">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>
 <b>by </b><i><a class="txt" href="#E4:142_1"><i><font color="Green" title="E89">A35</font></i></a>, <a class="ref" href="polynom1.html#T32" target="_self" title="POLYNOM1:th.32">Th32</a></i>;<br/><b>set </b><font color="Maroon" title="c13">ddbi1</font> =  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>;<br/><a NAME="E11:142_1"/><i><font color="Green" title="E93">A39</font></i>: 
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:142"><i><font color="Green" title="E88">A4</font></i></a>, <a class="txt" href="#E7:142_1"><i><font color="Green" title="E90">A36</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c14">b1</font>, <font color="Maroon" title="c15">b2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E13:142_1"/><i><font color="Green" title="E94">A40</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="c14">b1</font>,<font color="Maroon" title="c15">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E14:142_1"/><i><font color="Green" title="E95">A41</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c15">b2</font>
 <b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="#E7:142_1"><i><font color="Green" title="E90">A36</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">b1'</font> = <font color="Maroon" title="c14">b1</font>, <font color="Maroon" title="c17">b2'</font> = <font color="Maroon" title="c15">b2</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/><a NAME="E16:142_1"/><i><font color="Green" title="E96">A42</font></i>: 
( <font color="Maroon" title="c16">b1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b1</font> &amp; <font color="Maroon" title="c17">b2'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b2</font> )
 ;<br/><a NAME="E17:142_1"/><b>then </b><i><font color="Green" title="E97">A43</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b1</font>
 <b>by </b><i><a class="txt" href="#E13:142_1"><i><font color="Green" title="E94">A40</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><a NAME="E18:142_1"/><i><font color="Green" title="E98">A44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b2</font>
 <b>by </b><i><a class="txt" href="#E13:142_1"><i><font color="Green" title="E94">A40</font></i></a>, <a class="txt" href="#E16:142_1"><i><font color="Green" title="E96">A42</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><i><font color="Green" title="E99">A45</font></i>:  <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E11:142_1"><i><font color="Green" title="E93">A39</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>

;<br/><b>then consider </b><font color="Maroon" title="c18">b11</font>, <font color="Maroon" title="c19">b12</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E21:142_1"/><i><font color="Green" title="E100">A46</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="c18">b11</font>,<font color="Maroon" title="c19">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E22:142_1"/><i><font color="Green" title="E101">A47</font></i>: 
<font color="Maroon" title="c15">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b11</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c19">b12</font>
 <b>by </b><i><a class="txt" href="#E8:142_1"><i><font color="Green" title="E91">A37</font></i></a>, <a class="txt" href="#E18:142_1"><i><font color="Green" title="E98">A44</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/><a NAME="E23:142_1"/><i><font color="Green" title="E102">A48</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <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>
 <b>by </b><i><a class="txt" href="#E8:142_1"><i><font color="Green" title="E91">A37</font></i></a>, <a class="txt" href="#E19:142_1"><i><font color="Green" title="E99">A45</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E24:142_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E25:142_1"/><b>then </b>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&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="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E8:142_1"><i><font color="Green" title="E91">A37</font></i></a>, <a class="txt" href="#E19:142_1"><i><font color="Green" title="E99">A45</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>then </b><i><font color="Green" title="E103">A49</font></i>: <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="c14">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="c18">b11</font>,<font color="Maroon" title="c19">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E8:142_1"><i><font color="Green" title="E91">A37</font></i></a>, <a class="txt" href="#E11:142_1"><i><font color="Green" title="E93">A39</font></i></a>, <a class="txt" href="#E17:142_1"><i><font color="Green" title="E97">A43</font></i></a>, <a class="txt" href="#E21:142_1"><i><font color="Green" title="E100">A46</font></i></a>, <a class="txt" href="#E23:142_1"><i><font color="Green" title="E102">A48</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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">b1</font>,<font color="Maroon" title="c18">b11</font>,<font color="Maroon" title="c19">b12</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>
;<br/><a NAME="E27:142_1"/>
<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="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c18">b11</font></span>)</span> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c19">b12</font>
 <b>by </b><i><a class="txt" href="#E14:142_1"><i><font color="Green" title="E95">A41</font></i></a>, <a class="txt" href="#E22:142_1"><i><font color="Green" title="E101">A47</font></i></a>, <a class="ref" href="polynom1.html#T39" target="_self" title="POLYNOM1:th.39">Th39</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">i'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E29:142_1"/><i><font color="Green" title="E104">A50</font></i>: 
<font color="Maroon" title="c20">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> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>and </b><br/><a NAME="E30:142_1"/><i><font color="Green" title="E105">A51</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">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"><span class="p2">(<span class="default"><font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c18">b11</font></span>)</span>,<font color="Maroon" title="c19">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#T73" target="_self" title="POLYNOM1:th.73">Th73</a></i>;<br/><b>set </b><font color="Maroon" title="c21">b3</font> = <font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c18">b11</font>;<br/><b>consider </b><font color="Maroon" title="c22">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> <b> such that </b><br/><a NAME="E32:142_1"/><i><font color="Green" title="E106">A52</font></i>: 
<font color="Maroon" title="c22">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"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c18">b11</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E33:142_1"/><i><font color="Green" title="E107">A53</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c18">b11</font></span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c22">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="c14">b1</font>,<font color="Maroon" title="c18">b11</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#T73" target="_self" title="POLYNOM1:th.73">Th73</a></i>;<br/><b>set </b><font color="Maroon" title="c23">ddbi'2</font> =  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>;<br/><a NAME="E34:142_1"/><i><font color="Green" title="E108">A54</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E3:142"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="#E29:142_1"><i><font color="Green" title="E104">A50</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">b3'</font> = <font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c18">b11</font>, <font color="Maroon" title="c25">b12'</font> = <font color="Maroon" title="c19">b12</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/><a NAME="E36:142_1"/><i><font color="Green" title="E109">A55</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c24">b3'</font>,<font color="Maroon" title="c25">b12'</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E30:142_1"><i><font color="Green" title="E105">A51</font></i></a></i>;<br/><a NAME="E37:142_1"/><b>then </b><i><font color="Green" title="E110">A56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b12</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><a NAME="E38:142_1"/><i><font color="Green" title="E111">A57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c18">b11</font>
 <b>by </b><i><a class="txt" href="#E36:142_1"><i><font color="Green" title="E109">A55</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><i><font color="Green" title="E112">A58</font></i>:  <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">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"><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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E34:142_1"><i><font color="Green" title="E108">A54</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>

;<br/><a NAME="E40:142_1"/><i><font color="Green" title="E113">A59</font></i>: 
<font color="Maroon" title="c22">j'</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E32:142_1"><i><font color="Green" title="E106">A52</font></i></a>, <a class="txt" href="#E38:142_1"><i><font color="Green" title="E111">A57</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E41:142_1"/><i><font color="Green" title="E114">A60</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c18">b11</font></span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c22">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c18">b11</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j'</font>
 <b>by </b><i><a class="txt" href="#E32:142_1"><i><font color="Green" title="E106">A52</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><i><font color="Green" title="E115">A61</font></i>: <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">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <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="c22">j'</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c20">i'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E32:142_1"><i><font color="Green" title="E106">A52</font></i></a>, <a class="txt" href="#E34:142_1"><i><font color="Green" title="E108">A54</font></i></a>, <a class="txt" href="#E38:142_1"><i><font color="Green" title="E111">A57</font></i></a>, <a class="txt" href="#E39:142_1"><i><font color="Green" title="E112">A58</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c18">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="c19">b12</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E33:142_1"><i><font color="Green" title="E107">A53</font></i></a>, <a class="txt" href="#E37:142_1"><i><font color="Green" title="E110">A56</font></i></a>, <a class="txt" href="#E38:142_1"><i><font color="Green" title="E111">A57</font></i></a>, <a class="txt" href="#E40:142_1"><i><font color="Green" title="E113">A59</font></i></a>, <a class="txt" href="#E41:142_1"><i><font color="Green" title="E114">A60</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<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">b1</font>,<font color="Maroon" title="c18">b11</font>,<font color="Maroon" title="c19">b12</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>
;<br/><b>set </b><font color="Maroon" title="c26">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="polynom1.html#K4" title="POLYNOM1:func.4">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">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="c22">j'</font>;<br/><a NAME="E43:142_1"/><i><font color="Green" title="E116">A62</font></i>: 
<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="polynom1.html#K4" title="POLYNOM1:func.4">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">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="c22">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"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E29:142_1"><i><font color="Green" title="E104">A50</font></i></a>, <a class="txt" href="#E32:142_1"><i><font color="Green" title="E106">A52</font></i></a>, <a class="txt" href="#E38:142_1"><i><font color="Green" title="E111">A57</font></i></a>, <a class="txt" href="#E39:142_1"><i><font color="Green" title="E112">A58</font></i></a>, <a class="ref" href="polynom1.html#T33" target="_self" title="POLYNOM1:th.33">Th33</a></i>;<br/><a NAME="E44:142_1"/>
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="polynom1.html#K4" title="POLYNOM1:func.4">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="c20">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="c22">j'</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="c20">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j'</font>
 <b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E29:142_1"><i><font color="Green" title="E104">A50</font></i></a>, <a class="txt" href="#E32:142_1"><i><font color="Green" title="E106">A52</font></i></a>, <a class="txt" href="#E38:142_1"><i><font color="Green" title="E111">A57</font></i></a>, <a class="txt" href="#E39:142_1"><i><font color="Green" title="E112">A58</font></i></a>, <a class="ref" href="polynom1.html#T33" target="_self" title="POLYNOM1:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E45:142_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:142_1"><i><font color="Green" title="E89">A35</font></i></a>, <a class="txt" href="#E10:142_1"><i><font color="Green" title="E92">A38</font></i></a>, <a class="txt" href="#E26:142_1"><i><font color="Green" title="E103">A49</font></i></a>, <a class="txt" href="#E42:142_1"><i><font color="Green" title="E115">A61</font></i></a>, <a class="txt" href="#E43:142_1"><i><font color="Green" title="E116">A62</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:142"/><b>then </b><i><font color="Green" title="E89">A63</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E7:142"/><i><font color="Green" title="E90">A64</font></i>: 
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="142_2"><b>proof </b></a><div class="add">

<b>let </b><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="dtconstr.html#K15" title="DTCONSTR:func.15">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/>

<b>assume </b><b>that </b><br/><a NAME="E1:142_2"/><i><font color="Green" title="E91">A65</font></i>: 
<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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:142_2"/><i><font color="Green" title="E92">A66</font></i>: 
<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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E3:142_2"/><i><font color="Green" title="E93">A67</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="dtconstr.html#K15" title="DTCONSTR:func.15">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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>consider </b><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> <b> such that </b><br/><a NAME="E5:142_2"/><i><font color="Green" title="E94">A68</font></i>: 
<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>
 <b>and </b><br/><a NAME="E6:142_2"/><i><font color="Green" title="E95">A69</font></i>: 
<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>
 <b>and </b><br/><a NAME="E7:142_2"/><i><font color="Green" title="E96">A70</font></i>: 
<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="polynom1.html#K4" title="POLYNOM1:func.4">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>
 <b>and </b><br/><a NAME="E8:142_2"/><i><font color="Green" title="E97">A71</font></i>: 
<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="dtconstr.html#K15" title="DTCONSTR:func.15">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>
 <b>by </b><i><a class="txt" href="#E1:142_2"><i><font color="Green" title="E91">A65</font></i></a>, <a class="ref" href="polynom1.html#T32" target="_self" title="POLYNOM1:th.32">Th32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">i2</font>, <font color="Maroon" title="c13">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> <b> such that </b><br/><a NAME="E10:142_2"/><i><font color="Green" title="E98">A72</font></i>: 
<font color="Maroon" title="c12">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>
 <b>and </b><br/><a NAME="E11:142_2"/><i><font color="Green" title="E99">A73</font></i>: 
<font color="Maroon" title="c13">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="c12">i2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:142_2"/><i><font color="Green" title="E100">A74</font></i>: 
<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="polynom1.html#K4" title="POLYNOM1:func.4">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="c12">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="c13">j2</font>
 <b>and </b><br/><a NAME="E13:142_2"/><i><font color="Green" title="E101">A75</font></i>: 
<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="c12">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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>
 <b>by </b><i><a class="txt" href="#E2:142_2"><i><font color="Green" title="E92">A66</font></i></a>, <a class="ref" href="polynom1.html#T32" target="_self" title="POLYNOM1:th.32">Th32</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">ddbi11</font> =  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>;<br/>
<a NAME="E14:142_2"/><i><font color="Green" title="E102">A76</font></i>: 
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:142"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="#E5:142_2"><i><font color="Green" title="E94">A68</font></i></a></i>;<br/>
<a NAME="E15:142_2"/><i><font color="Green" title="E103">A77</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="polynom1.html#K19" title="POLYNOM1:func.19">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>
 
<b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E5:142_2"><i><font color="Green" title="E94">A68</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">b11</font>, <font color="Maroon" title="c16">b12</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E17:142_2"/><i><font color="Green" title="E104">A78</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="c15">b11</font>,<font color="Maroon" title="c16">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E18:142_2"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b11</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c16">b12</font>
 <b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E5:142_2"><i><font color="Green" title="E94">A68</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">b11'</font> = <font color="Maroon" title="c15">b11</font>, <font color="Maroon" title="c18">b12'</font> = <font color="Maroon" title="c16">b12</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/>
<a NAME="E20:142_2"/><i><font color="Green" title="E105">A79</font></i>: 
( <font color="Maroon" title="c17">b11'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b11</font> &amp; <font color="Maroon" title="c18">b12'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">b12</font> )
 
;<br/>
<a NAME="E21:142_2"/><b>then </b><i><font color="Green" title="E106">A80</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b11</font>
 
<b>by </b><i><a class="txt" href="#E17:142_2"><i><font color="Green" title="E104">A78</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E22:142_2"/><i><font color="Green" title="E107">A81</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b12</font>
 
<b>by </b><i><a class="txt" href="#E17:142_2"><i><font color="Green" title="E104">A78</font></i></a>, <a class="txt" href="#E20:142_2"><i><font color="Green" title="E105">A79</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<i><font color="Green" title="E108">A82</font></i>:  <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E14:142_2"><i><font color="Green" title="E102">A76</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>

;<br/>
<b>then consider </b><font color="Maroon" title="c19">b111</font>, <font color="Maroon" title="c20">b112</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E25:142_2"/><i><font color="Green" title="E109">A83</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="c19">b111</font>,<font color="Maroon" title="c20">b112</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E26:142_2"/><i><font color="Green" title="E110">A84</font></i>: 
<font color="Maroon" title="c15">b11</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">b111</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c20">b112</font>
 <b>by </b><i><a class="txt" href="#E6:142_2"><i><font color="Green" title="E95">A69</font></i></a>, <a class="txt" href="#E21:142_2"><i><font color="Green" title="E106">A80</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/>
<a NAME="E27:142_2"/><i><font color="Green" title="E111">A85</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <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>
 
<b>by </b><i><a class="txt" href="#E6:142_2"><i><font color="Green" title="E95">A69</font></i></a>, <a class="txt" href="#E23:142_2"><i><font color="Green" title="E108">A82</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E28:142_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E29:142_2"/><b>then </b>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&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="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E6:142_2"><i><font color="Green" title="E95">A69</font></i></a>, <a class="txt" href="#E23:142_2"><i><font color="Green" title="E108">A82</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>then </b><i><font color="Green" title="E112">A86</font></i>: <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="c19">b111</font>,<font color="Maroon" title="c20">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="c16">b12</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E6:142_2"><i><font color="Green" title="E95">A69</font></i></a>, <a class="txt" href="#E14:142_2"><i><font color="Green" title="E102">A76</font></i></a>, <a class="txt" href="#E22:142_2"><i><font color="Green" title="E107">A81</font></i></a>, <a class="txt" href="#E25:142_2"><i><font color="Green" title="E109">A83</font></i></a>, <a class="txt" href="#E27:142_2"><i><font color="Green" title="E111">A85</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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">b111</font>,<font color="Maroon" title="c20">b112</font>,<font color="Maroon" title="c16">b12</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c21">ddbi21</font> =  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>;<br/>
<a NAME="E31:142_2"/><i><font color="Green" title="E113">A87</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:142"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="#E10:142_2"><i><font color="Green" title="E98">A72</font></i></a></i>;<br/>
<a NAME="E32:142_2"/><i><font color="Green" title="E114">A88</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c12">i2</font>
 
<b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E10:142_2"><i><font color="Green" title="E98">A72</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">b21</font>, <font color="Maroon" title="c23">b22</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E34:142_2"/><i><font color="Green" title="E115">A89</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">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="c22">b21</font>,<font color="Maroon" title="c23">b22</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E35:142_2"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">b21</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c23">b22</font>
 <b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E10:142_2"><i><font color="Green" title="E98">A72</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">b21'</font> = <font color="Maroon" title="c22">b21</font>, <font color="Maroon" title="c25">b22'</font> = <font color="Maroon" title="c23">b22</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/>
<a NAME="E37:142_2"/><i><font color="Green" title="E116">A90</font></i>: 
( <font color="Maroon" title="c24">b21'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">b21</font> &amp; <font color="Maroon" title="c25">b22'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">b22</font> )
 
;<br/>
<a NAME="E38:142_2"/><b>then </b><i><font color="Green" title="E117">A91</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">b21</font>
 
<b>by </b><i><a class="txt" href="#E34:142_2"><i><font color="Green" title="E115">A89</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E39:142_2"/><i><font color="Green" title="E118">A92</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b22</font>
 
<b>by </b><i><a class="txt" href="#E34:142_2"><i><font color="Green" title="E115">A89</font></i></a>, <a class="txt" href="#E37:142_2"><i><font color="Green" title="E116">A90</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<i><font color="Green" title="E119">A93</font></i>:  <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="c12">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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E31:142_2"><i><font color="Green" title="E113">A87</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>

;<br/>
<b>then consider </b><font color="Maroon" title="c26">b211</font>, <font color="Maroon" title="c27">b212</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E42:142_2"/><i><font color="Green" title="E120">A94</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c13">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="c26">b211</font>,<font color="Maroon" title="c27">b212</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E43:142_2"/><i><font color="Green" title="E121">A95</font></i>: 
<font color="Maroon" title="c22">b21</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">b211</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c27">b212</font>
 <b>by </b><i><a class="txt" href="#E11:142_2"><i><font color="Green" title="E99">A73</font></i></a>, <a class="txt" href="#E38:142_2"><i><font color="Green" title="E117">A91</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/>
<a NAME="E44:142_2"/><i><font color="Green" title="E122">A96</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E45:142_2"/><i><font color="Green" title="E123">A97</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <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="c13">j2</font>
 
<b>by </b><i><a class="txt" href="#E11:142_2"><i><font color="Green" title="E99">A73</font></i></a>, <a class="txt" href="#E40:142_2"><i><font color="Green" title="E119">A93</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E46:142_2"/>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E11:142_2"><i><font color="Green" title="E99">A73</font></i></a>, <a class="txt" href="#E40:142_2"><i><font color="Green" title="E119">A93</font></i></a>, <a class="txt" href="#E44:142_2"><i><font color="Green" title="E122">A96</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>then </b><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="c12">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">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="c26">b211</font>,<font color="Maroon" title="c27">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="c23">b22</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E11:142_2"><i><font color="Green" title="E99">A73</font></i></a>, <a class="txt" href="#E31:142_2"><i><font color="Green" title="E113">A87</font></i></a>, <a class="txt" href="#E39:142_2"><i><font color="Green" title="E118">A92</font></i></a>, <a class="txt" href="#E42:142_2"><i><font color="Green" title="E120">A94</font></i></a>, <a class="txt" href="#E45:142_2"><i><font color="Green" title="E123">A97</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c26">b211</font>,<font color="Maroon" title="c27">b212</font>,<font color="Maroon" title="c23">b22</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>
;<br/>
<a NAME="E48:142_2"/><b>then </b><i><font color="Green" title="E124">A98</font></i>: 
( <font color="Maroon" title="c19">b111</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">b211</font> &amp; <font color="Maroon" title="c20">b112</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c27">b212</font> &amp; <font color="Maroon" title="c16">b12</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">b22</font> )
 
<b>by </b><i><a class="txt" href="#E3:142_2"><i><font color="Green" title="E93">A67</font></i></a>, <a class="txt" href="#E8:142_2"><i><font color="Green" title="E97">A71</font></i></a>, <a class="txt" href="#E13:142_2"><i><font color="Green" title="E101">A75</font></i></a>, <a class="txt" href="#E30:142_2"><i><font color="Green" title="E112">A86</font></i></a>, <a class="ref" href="group_7.html#T3" title="GROUP_7:th.3">GROUP_7:3</a></i>;<br/>
<a NAME="E49:142_2"/><b>then </b>
<font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font>
 
<b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="#E5:142_2"><i><font color="Green" title="E94">A68</font></i></a>, <a class="txt" href="#E10:142_2"><i><font color="Green" title="E98">A72</font></i></a>, <a class="txt" href="#E15:142_2"><i><font color="Green" title="E103">A77</font></i></a>, <a class="txt" href="#E17:142_2"><i><font color="Green" title="E104">A78</font></i></a>, <a class="txt" href="#E26:142_2"><i><font color="Green" title="E110">A84</font></i></a>, <a class="txt" href="#E32:142_2"><i><font color="Green" title="E114">A88</font></i></a>, <a class="txt" href="#E34:142_2"><i><font color="Green" title="E115">A89</font></i></a>, <a class="txt" href="#E43:142_2"><i><font color="Green" title="E121">A95</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E50:142_2"/>
<font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font>
 <b>by </b><i><a class="txt" href="#E6:142_2"><i><font color="Green" title="E95">A69</font></i></a>, <a class="txt" href="#E7:142_2"><i><font color="Green" title="E96">A70</font></i></a>, <a class="txt" href="#E11:142_2"><i><font color="Green" title="E99">A73</font></i></a>, <a class="txt" href="#E12:142_2"><i><font color="Green" title="E100">A74</font></i></a>, <a class="txt" href="#E25:142_2"><i><font color="Green" title="E109">A83</font></i></a>, <a class="txt" href="#E27:142_2"><i><font color="Green" title="E111">A85</font></i></a>, <a class="txt" href="#E40:142_2"><i><font color="Green" title="E119">A93</font></i></a>, <a class="txt" href="#E42:142_2"><i><font color="Green" title="E120">A94</font></i></a>, <a class="txt" href="#E45:142_2"><i><font color="Green" title="E123">A97</font></i></a>, <a class="txt" href="#E48:142_2"><i><font color="Green" title="E124">A98</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:142"/>
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="142_3"><b>proof </b></a><div class="add">

<b>let </b><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="dtconstr.html#K15" title="DTCONSTR:func.15">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/>

<b>assume </b><b>that </b><br/><a NAME="E1:142_3"/><i><font color="Green" title="E91">A99</font></i>: 
<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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <b>and </b><br/><a NAME="E2:142_3"/><i><font color="Green" title="E92">A100</font></i>: 
<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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <b>and </b><br/><a NAME="E3:142_3"/><i><font color="Green" title="E93">A101</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="dtconstr.html#K15" title="DTCONSTR:func.15">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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>consider </b><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> <b> such that </b><br/><a NAME="E5:142_3"/><i><font color="Green" title="E94">A102</font></i>: 
<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>
 <b>and </b><br/><a NAME="E6:142_3"/><i><font color="Green" title="E95">A103</font></i>: 
<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>
 <b>and </b><br/><a NAME="E7:142_3"/><i><font color="Green" title="E96">A104</font></i>: 
<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="polynom1.html#K4" title="POLYNOM1:func.4">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>
 <b>and </b><br/><a NAME="E8:142_3"/><i><font color="Green" title="E97">A105</font></i>: 
<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="dtconstr.html#K15" title="DTCONSTR:func.15">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>
 <b>by </b><i><a class="txt" href="#E1:142_3"><i><font color="Green" title="E91">A99</font></i></a>, <a class="ref" href="polynom1.html#T32" target="_self" title="POLYNOM1:th.32">Th32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">i2</font>, <font color="Maroon" title="c13">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> <b> such that </b><br/><a NAME="E10:142_3"/><i><font color="Green" title="E98">A106</font></i>: 
<font color="Maroon" title="c12">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>
 <b>and </b><br/><a NAME="E11:142_3"/><i><font color="Green" title="E99">A107</font></i>: 
<font color="Maroon" title="c13">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="c12">i2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:142_3"/><i><font color="Green" title="E100">A108</font></i>: 
<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="polynom1.html#K4" title="POLYNOM1:func.4">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="c12">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="c13">j2</font>
 <b>and </b><br/><a NAME="E13:142_3"/><i><font color="Green" title="E101">A109</font></i>: 
<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="c12">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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>
 <b>by </b><i><a class="txt" href="#E2:142_3"><i><font color="Green" title="E92">A100</font></i></a>, <a class="ref" href="polynom1.html#T32" target="_self" title="POLYNOM1:th.32">Th32</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">ddbi11</font> =  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>;<br/>
<a NAME="E14:142_3"/><i><font color="Green" title="E102">A110</font></i>: 
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:142"><i><font color="Green" title="E88">A4</font></i></a>, <a class="txt" href="#E5:142_3"><i><font color="Green" title="E94">A102</font></i></a></i>;<br/>
<a NAME="E15:142_3"/><i><font color="Green" title="E103">A111</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="polynom1.html#K19" title="POLYNOM1:func.19">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>
 
<b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="#E5:142_3"><i><font color="Green" title="E94">A102</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">b11</font>, <font color="Maroon" title="c16">b12</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E17:142_3"/><i><font color="Green" title="E104">A112</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="c15">b11</font>,<font color="Maroon" title="c16">b12</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E18:142_3"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b11</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c16">b12</font>
 <b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="#E5:142_3"><i><font color="Green" title="E94">A102</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">b11'</font> = <font color="Maroon" title="c15">b11</font>, <font color="Maroon" title="c18">b12'</font> = <font color="Maroon" title="c16">b12</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/>
<a NAME="E20:142_3"/><i><font color="Green" title="E105">A113</font></i>: 
( <font color="Maroon" title="c17">b11'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b11</font> &amp; <font color="Maroon" title="c18">b12'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">b12</font> )
 
;<br/>
<a NAME="E21:142_3"/><b>then </b><i><font color="Green" title="E106">A114</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b11</font>
 
<b>by </b><i><a class="txt" href="#E17:142_3"><i><font color="Green" title="E104">A112</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E22:142_3"/><i><font color="Green" title="E107">A115</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b12</font>
 
<b>by </b><i><a class="txt" href="#E17:142_3"><i><font color="Green" title="E104">A112</font></i></a>, <a class="txt" href="#E20:142_3"><i><font color="Green" title="E105">A113</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<i><font color="Green" title="E108">A116</font></i>:  <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E14:142_3"><i><font color="Green" title="E102">A110</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>

;<br/>
<b>then consider </b><font color="Maroon" title="c19">b111</font>, <font color="Maroon" title="c20">b112</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E25:142_3"/><i><font color="Green" title="E109">A117</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="c19">b111</font>,<font color="Maroon" title="c20">b112</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E26:142_3"/><i><font color="Green" title="E110">A118</font></i>: 
<font color="Maroon" title="c16">b12</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">b111</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c20">b112</font>
 <b>by </b><i><a class="txt" href="#E6:142_3"><i><font color="Green" title="E95">A103</font></i></a>, <a class="txt" href="#E22:142_3"><i><font color="Green" title="E107">A115</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/>
<a NAME="E27:142_3"/><i><font color="Green" title="E111">A119</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E28:142_3"/><i><font color="Green" title="E112">A120</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <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>
 
<b>by </b><i><a class="txt" href="#E6:142_3"><i><font color="Green" title="E95">A103</font></i></a>, <a class="txt" href="#E23:142_3"><i><font color="Green" title="E108">A116</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E29:142_3"/>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&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="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c10">i1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E6:142_3"><i><font color="Green" title="E95">A103</font></i></a>, <a class="txt" href="#E23:142_3"><i><font color="Green" title="E108">A116</font></i></a>, <a class="txt" href="#E27:142_3"><i><font color="Green" title="E111">A119</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>then </b><i><font color="Green" title="E113">A121</font></i>: <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="c15">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="c19">b111</font>,<font color="Maroon" title="c20">b112</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E6:142_3"><i><font color="Green" title="E95">A103</font></i></a>, <a class="txt" href="#E14:142_3"><i><font color="Green" title="E102">A110</font></i></a>, <a class="txt" href="#E21:142_3"><i><font color="Green" title="E106">A114</font></i></a>, <a class="txt" href="#E25:142_3"><i><font color="Green" title="E109">A117</font></i></a>, <a class="txt" href="#E28:142_3"><i><font color="Green" title="E112">A120</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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">b11</font>,<font color="Maroon" title="c19">b111</font>,<font color="Maroon" title="c20">b112</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c21">ddbi21</font> =  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>;<br/>
<a NAME="E31:142_3"/><i><font color="Green" title="E114">A122</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:142"><i><font color="Green" title="E88">A4</font></i></a>, <a class="txt" href="#E10:142_3"><i><font color="Green" title="E98">A106</font></i></a></i>;<br/>
<a NAME="E32:142_3"/><i><font color="Green" title="E115">A123</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c12">i2</font>
 
<b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="#E10:142_3"><i><font color="Green" title="E98">A106</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">b21</font>, <font color="Maroon" title="c23">b22</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E34:142_3"/><i><font color="Green" title="E116">A124</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">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="c22">b21</font>,<font color="Maroon" title="c23">b22</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E35:142_3"/>
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">b21</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c23">b22</font>
 <b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="#E10:142_3"><i><font color="Green" title="E98">A106</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">b21'</font> = <font color="Maroon" title="c22">b21</font>, <font color="Maroon" title="c25">b22'</font> = <font color="Maroon" title="c23">b22</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/>
<a NAME="E37:142_3"/><i><font color="Green" title="E117">A125</font></i>: 
( <font color="Maroon" title="c24">b21'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">b21</font> &amp; <font color="Maroon" title="c25">b22'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">b22</font> )
 
;<br/>
<a NAME="E38:142_3"/><b>then </b><i><font color="Green" title="E118">A126</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">b21</font>
 
<b>by </b><i><a class="txt" href="#E34:142_3"><i><font color="Green" title="E116">A124</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E39:142_3"/><i><font color="Green" title="E119">A127</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b22</font>
 
<b>by </b><i><a class="txt" href="#E34:142_3"><i><font color="Green" title="E116">A124</font></i></a>, <a class="txt" href="#E37:142_3"><i><font color="Green" title="E117">A125</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<i><font color="Green" title="E120">A128</font></i>:  <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="c12">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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E31:142_3"><i><font color="Green" title="E114">A122</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>

;<br/>
<b>then consider </b><font color="Maroon" title="c26">b211</font>, <font color="Maroon" title="c27">b212</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E42:142_3"/><i><font color="Green" title="E121">A129</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c13">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="c26">b211</font>,<font color="Maroon" title="c27">b212</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E43:142_3"/><i><font color="Green" title="E122">A130</font></i>: 
<font color="Maroon" title="c23">b22</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">b211</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c27">b212</font>
 <b>by </b><i><a class="txt" href="#E11:142_3"><i><font color="Green" title="E99">A107</font></i></a>, <a class="txt" href="#E39:142_3"><i><font color="Green" title="E119">A127</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/>
<a NAME="E44:142_3"/><i><font color="Green" title="E123">A131</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <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="c13">j2</font>
 
<b>by </b><i><a class="txt" href="#E11:142_3"><i><font color="Green" title="E99">A107</font></i></a>, <a class="txt" href="#E40:142_3"><i><font color="Green" title="E120">A128</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E45:142_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E46:142_3"/><b>then </b>
<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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&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="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom1.html#K2" title="POLYNOM1:func.2">/.</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E11:142_3"><i><font color="Green" title="E99">A107</font></i></a>, <a class="txt" href="#E40:142_3"><i><font color="Green" title="E120">A128</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>then </b><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="c12">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">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="c22">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="c26">b211</font>,<font color="Maroon" title="c27">b212</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E11:142_3"><i><font color="Green" title="E99">A107</font></i></a>, <a class="txt" href="#E31:142_3"><i><font color="Green" title="E114">A122</font></i></a>, <a class="txt" href="#E38:142_3"><i><font color="Green" title="E118">A126</font></i></a>, <a class="txt" href="#E42:142_3"><i><font color="Green" title="E121">A129</font></i></a>, <a class="txt" href="#E44:142_3"><i><font color="Green" title="E123">A131</font></i></a>, <a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c22">b21</font>,<font color="Maroon" title="c26">b211</font>,<font color="Maroon" title="c27">b212</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>
;<br/>
<a NAME="E48:142_3"/><b>then </b><i><font color="Green" title="E124">A132</font></i>: 
( <font color="Maroon" title="c19">b111</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">b211</font> &amp; <font color="Maroon" title="c20">b112</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c27">b212</font> &amp; <font color="Maroon" title="c15">b11</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">b21</font> )
 
<b>by </b><i><a class="txt" href="#E3:142_3"><i><font color="Green" title="E93">A101</font></i></a>, <a class="txt" href="#E8:142_3"><i><font color="Green" title="E97">A105</font></i></a>, <a class="txt" href="#E13:142_3"><i><font color="Green" title="E101">A109</font></i></a>, <a class="txt" href="#E30:142_3"><i><font color="Green" title="E113">A121</font></i></a>, <a class="ref" href="group_7.html#T3" title="GROUP_7:th.3">GROUP_7:3</a></i>;<br/>
<a NAME="E49:142_3"/><b>then </b>
<font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font>
 
<b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="#E5:142_3"><i><font color="Green" title="E94">A102</font></i></a>, <a class="txt" href="#E10:142_3"><i><font color="Green" title="E98">A106</font></i></a>, <a class="txt" href="#E15:142_3"><i><font color="Green" title="E103">A111</font></i></a>, <a class="txt" href="#E17:142_3"><i><font color="Green" title="E104">A112</font></i></a>, <a class="txt" href="#E26:142_3"><i><font color="Green" title="E110">A118</font></i></a>, <a class="txt" href="#E32:142_3"><i><font color="Green" title="E115">A123</font></i></a>, <a class="txt" href="#E34:142_3"><i><font color="Green" title="E116">A124</font></i></a>, <a class="txt" href="#E43:142_3"><i><font color="Green" title="E122">A130</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E50:142_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>
 <b>by </b><i><a class="txt" href="#E6:142_3"><i><font color="Green" title="E95">A103</font></i></a>, <a class="txt" href="#E7:142_3"><i><font color="Green" title="E96">A104</font></i></a>, <a class="txt" href="#E11:142_3"><i><font color="Green" title="E99">A107</font></i></a>, <a class="txt" href="#E12:142_3"><i><font color="Green" title="E100">A108</font></i></a>, <a class="txt" href="#E25:142_3"><i><font color="Green" title="E109">A117</font></i></a>, <a class="txt" href="#E28:142_3"><i><font color="Green" title="E112">A120</font></i></a>, <a class="txt" href="#E40:142_3"><i><font color="Green" title="E120">A128</font></i></a>, <a class="txt" href="#E42:142_3"><i><font color="Green" title="E121">A129</font></i></a>, <a class="txt" href="#E44:142_3"><i><font color="Green" title="E123">A131</font></i></a>, <a class="txt" href="#E48:142_3"><i><font color="Green" title="E124">A132</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:142"/><b>then </b>
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font>, <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c4">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="#E6:142"><i><font color="Green" title="E89">A63</font></i></a>, <a class="txt" href="#E7:142"><i><font color="Green" title="E90">A64</font></i></a>, <a class="ref" href="rfinseq.html#T39" title="RFINSEQ:th.39">RFINSEQ:39</a></i>;<br/>
<b>hence </b><a NAME="E10:142"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> st  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="dtconstr.html#K15" title="DTCONSTR:func.15">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>
 <b>by </b><i><a class="ref" href="rfinseq.html#T17" title="RFINSEQ:th.17">RFINSEQ:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
