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<b>let </b><font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c2">o2</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">b1</font> being   <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">o1</font><br/> for <font color="Olive" title="b2">b2</font> being   <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="c2">o2</font><br/> for <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">2 <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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></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="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">b1</font></span>)</span> &amp; ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">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="Olive" title="b1">b1</font></span>)</span> holds <br/> ex <font color="Olive" title="b5">a1'</font>, <font color="Olive" title="b6">b1'</font> being   <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">o1</font> ex <font color="Olive" title="b7">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b7">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b4">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b5">a1'</font>,<font color="Olive" title="b6">b1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b7">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b2">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b8">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b8">m</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="b7">Fk</font> holds <br/> ex <font color="Olive" title="b9">a1''</font>, <font color="Olive" title="b10">b1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b9">a1''</font>,<font color="Olive" title="b10">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b7">Fk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b9">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b6">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b10">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) ) ) holds <br/> <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b3">G</font></span><br/><b>let </b><font color="Maroon" title="c3">b1</font> be    <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">o1</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">b2</font> being   <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="c2">o2</font><br/> for <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">2 <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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></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">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="c3">b1</font></span>)</span> &amp; ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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="c3">b1</font></span>)</span> holds <br/> ex <font color="Olive" title="b4">a1'</font>, <font color="Olive" title="b5">b1'</font> being   <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">o1</font> ex <font color="Olive" title="b6">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b6">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b3">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b4">a1'</font>,<font color="Olive" title="b5">b1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b1">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b7">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b7">m</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="b6">Fk</font> holds <br/> ex <font color="Olive" title="b8">a1''</font>, <font color="Olive" title="b9">b1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Olive" title="b1">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b8">a1''</font>,<font color="Olive" title="b9">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b6">Fk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b8">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b9">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) ) ) holds <br/> <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b2">G</font></span><br/><b>let </b><font color="Maroon" title="c4">b2</font> be    <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="c2">o2</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">G</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default">2 <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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></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">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="c3">b1</font></span>)</span> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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="c3">b1</font></span>)</span> holds <br/> ex <font color="Olive" title="b3">a1'</font>, <font color="Olive" title="b4">b1'</font> being   <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">o1</font> ex <font color="Olive" title="b5">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b5">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b2">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b3">a1'</font>,<font color="Olive" title="b4">b1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b6">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">m</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="b5">Fk</font> holds <br/> ex <font color="Olive" title="b7">a1''</font>, <font color="Olive" title="b8">b1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b7">a1''</font>,<font color="Olive" title="b8">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b5">Fk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b7">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b8">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) ) ) holds <br/> <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b1">G</font></span><br/><b>let </b><font color="Maroon" title="c5">G</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default">2 <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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></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="c5">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="c3">b1</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c3">b1</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">a1'</font>, <font color="Olive" title="b3">b1'</font> being   <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">o1</font> ex <font color="Olive" title="b4">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b4">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">a1'</font>,<font color="Olive" title="b3">b1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b5">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">m</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="b4">Fk</font> holds <br/> ex <font color="Olive" title="b6">a1''</font>, <font color="Olive" title="b7">b1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b6">a1''</font>,<font color="Olive" title="b7">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b4">Fk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b6">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b7">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) ) ) implies  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">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="c3">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green" title="E17">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c3">b1</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">a1'</font>, <font color="Olive" title="b3">b1'</font> being   <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">o1</font> ex <font color="Olive" title="b4">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b4">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">a1'</font>,<font color="Olive" title="b3">b1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b5">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">m</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="b4">Fk</font> holds <br/> ex <font color="Olive" title="b6">a1''</font>, <font color="Olive" title="b7">b1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b6">a1''</font>,<font color="Olive" title="b7">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b4">Fk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b6">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b7">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>] <b>means </b>(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> $2 <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"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</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> $2 holds <br/> ex <font color="Olive" title="b2">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 ) ) );<br/>
<a NAME="E3:20"/><i><font color="Green" title="E18">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c5">G</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">k</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> <font color="Maroon" title="c5">G</font></span>)</span> implies  ex <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Olive" title="b1">p</font>] )</span><br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c6">k</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> <font color="Maroon" title="c5">G</font></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="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Olive" title="b1">p</font>]</span><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">q</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 );<br/>
<a NAME="E2:20_1"/><i><font color="Green" title="E20">A5</font></i>: 
for <font color="Olive" title="b1">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>   st <font color="Olive" title="b1">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"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">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"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">j</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c7">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"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></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">x</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">j</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:20_1_1"/>
<font color="Maroon" title="c6">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="c5">G</font>
 
<b>by </b><i><a class="txt" href="#E1:20_1"><i><font color="Green" title="E19">A4</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="E3:20_1_1"/><b>then </b><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:20_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">2 <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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E5:20_1_1"/><b>then </b>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="txt" href="#E3:20_1_1"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">Gk</font> = <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> ;<br/>
<a NAME="E7:20_1_1"/>
<font color="Maroon" title="c7">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="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:20_1_1"><i><font color="Green" title="E21">A6</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="E8:20_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Gk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c10">q1</font>, <font color="Maroon" title="c11">q2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span><b> such that </b><br/><a NAME="E11:20_1_1"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c11">q2</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T120" title="FINSEQ_2:th.120">FINSEQ_2:120</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <b>by </b><i><a class="txt" href="#E11:20_1_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c12">x</font>]</span><br/>

<b>thus </b><a NAME="E13:20_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c12">x</font>]
 ; <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>consider </b><font color="Maroon" title="c7">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green" title="E21">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> <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"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:20_1"/><i><font color="Green" title="E22">A10</font></i>: 
for <font color="Olive" title="b1">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>   st <font color="Olive" title="b1">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"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c7">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1" title="POLYNOM2:sch.1">POLYNOM2:sch 1</a>(<a class="txt" href="#E2:20_1"><i><font color="Green" title="E20">A5</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">p</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c8">p</font>]</span><br/>

<b>thus </b><a NAME="E7:20_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</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"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:20_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>; <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">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</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> <font color="Maroon" title="c8">p</font> holds <br/> ex <font color="Olive" title="b2">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )</span><br/>

<b>let </b><font color="Maroon" title="c9">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">j</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> <font color="Maroon" title="c8">p</font> implies  ex <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 ) )</span><br/>

<b>assume </b><a NAME="E8:20_1"/><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c9">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> <font color="Maroon" title="c8">p</font>
 ; <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="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )</span><br/>

<b>then consider </b><font color="Maroon" title="c10">q</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:20_1"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>and </b><br/><a NAME="E11:20_1"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="#E4:20_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="#E5:20_1"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )</span><br/>

<b>thus </b><a NAME="E12:20_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="#E10:20_1"><i><font color="Green" title="E24">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span><br/>

<b>thus </b><a NAME="E13:20_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="#E8:20_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="#E11:20_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1: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>
<b>consider </b><font color="Maroon" title="c6">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E5:20"/><i><font color="Green" title="E19">A14</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> <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> <font color="Maroon" title="c5">G</font></span>)</span>
 <b>and </b><br/><a NAME="E6:20"/><i><font color="Green" title="E20">A15</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="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> <font color="Maroon" title="c5">G</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1" title="POLYNOM2:sch.1">POLYNOM2:sch 1</a>(<a class="txt" href="#E3:20"><i><font color="Green" title="E18">A3</font></i></a>);<br/></i>
<a NAME="E7:20"/>
 <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="c3">b1</font></span>)</span> <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#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E8:20"/><b>then </b><i><font color="Green" title="E21">A16</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">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#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E5:20"><i><font color="Green" title="E19">A14</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="E9:20"/><i><font color="Green" title="E22">A17</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#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <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="c3">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E10:20"/><i><font color="Green" title="E23">A18</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">a1'</font> being   <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">o1</font> ex <font color="Olive" title="b3">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> st <br/>( <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">m</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">Fk</font> holds <br/> ex <font color="Olive" title="b5">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a1''</font> &amp; <font color="Olive" title="b3">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b5">a1''</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">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#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> implies  ex <font color="Olive" title="b1">a1'</font> being   <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">o1</font> ex <font color="Olive" title="b2">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> st <br/>( <font color="Olive" title="b2">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">m</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">Fk</font> holds <br/> ex <font color="Olive" title="b4">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a1''</font> &amp; <font color="Olive" title="b2">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b4">a1''</font> ) ) ) )</span><br/>

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green" title="E24">A19</font></i>: 
<font color="Maroon" title="c7">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#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></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">a1'</font> being   <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">o1</font> ex <font color="Olive" title="b2">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> st <br/>( <font color="Olive" title="b2">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">m</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">Fk</font> holds <br/> ex <font color="Olive" title="b4">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a1''</font> &amp; <font color="Olive" title="b2">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b4">a1''</font> ) ) )</span><br/>

<b>then consider </b><font color="Maroon" title="c8">a1'</font>, <font color="Maroon" title="c9">b1'</font> being    <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">o1</font>, <font color="Maroon" title="c10">Gi</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:20_2"/><i><font color="Green" title="E25">A20</font></i>: 
<font color="Maroon" title="c10">Gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font>
 <b>and </b><br/><a NAME="E4:20_2"/><i><font color="Green" title="E26">A21</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">a1'</font>,<font color="Maroon" title="c9">b1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E5:20_2"/><i><font color="Green" title="E27">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">Gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E6:20_2"/><i><font color="Green" title="E28">A23</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</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="c10">Gi</font> holds <br/> ex <font color="Olive" title="b2">a1''</font>, <font color="Olive" title="b3">b1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">a1''</font>,<font color="Olive" title="b3">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c10">Gi</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="#E2:20"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="#E9:20"><i><font color="Green" title="E22">A17</font></i></a></i>;<br/>
<a NAME="E7:20_2"/><i><font color="Green" title="E29">A24</font></i>: 
<font color="Maroon" title="c7">i</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> <font color="Maroon" title="c5">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E9:20"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="#E1:20_2"><i><font color="Green" title="E24">A19</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="E8:20_2"/><b>then </b><i><font color="Green" title="E30">A25</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="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <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"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 ) ) )
 
<b>by </b><i><a class="txt" href="#E6:20"><i><font color="Green" title="E20">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Fk</font> = <font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c8">a1'</font>
; <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">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> st <br/>( <font color="Olive" title="b1">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</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">Fk</font> holds <br/> ex <font color="Olive" title="b3">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a1''</font> &amp; <font color="Olive" title="b1">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">a1''</font> ) ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c11">Fk</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</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="c11">Fk</font> holds <br/> ex <font color="Olive" title="b2">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1''</font> &amp; <font color="Maroon" title="c11">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a1''</font> ) ) )</span><br/>

<a NAME="E10:20_2"/><i><font color="Green" title="E31">A26</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> <font color="Maroon" title="c4">b2</font></span>)</span> <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#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/>
<b>thus </b><a NAME="E11:20_2"/>
<font color="Maroon" title="c11">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</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="c11">Fk</font> holds <br/> ex <font color="Olive" title="b2">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1''</font> &amp; <font color="Maroon" title="c11">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a1''</font> ) ) )</span><br/>

<b>thus </b><a NAME="E12:20_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font>
 <b>by </b><i><a class="txt" href="#E9:20"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="#E1:20_2"><i><font color="Green" title="E24">A19</font></i></a>, <a class="txt" href="#E4:20_2"><i><font color="Green" title="E26">A21</font></i></a>, <a class="ref" href="polynom1.html#T74" title="POLYNOM1:th.74">POLYNOM1:74</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</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="c11">Fk</font> holds <br/> ex <font color="Olive" title="b2">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1''</font> &amp; <font color="Maroon" title="c11">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a1''</font> ) ) )</span><br/>

<b>thus </b> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Fk</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E8:20_2"><i><font color="Green" title="E30">A25</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E9:20"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="#E1:20_2"><i><font color="Green" title="E24">A19</font></i></a>, <a class="txt" href="#E3:20_2"><i><font color="Green" title="E25">A20</font></i></a>, <a class="txt" href="#E5:20_2"><i><font color="Green" title="E27">A22</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E10:20_2"><i><font color="Green" title="E31">A26</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</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="c11">Fk</font> holds <br/> ex <font color="Olive" title="b2">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1''</font> &amp; <font color="Maroon" title="c11">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a1''</font> )</span><br/>

<b>let </b><font color="Maroon" title="c12">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">m</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="c11">Fk</font> implies  ex <font color="Olive" title="b1">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1''</font> &amp; <font color="Maroon" title="c11">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">a1''</font> ) )</span><br/>

<a NAME="E14:20_2"/><i><font color="Green" title="E32">A27</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font>
 
<b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E9:20"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="#E1:20_2"><i><font color="Green" title="E24">A19</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>assume </b><a NAME="E15:20_2"/><i><font color="Green" title="E33">A28</font></i>: 
<font color="Maroon" title="c12">m</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="c11">Fk</font>
 ; <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">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1''</font> &amp; <font color="Maroon" title="c11">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">a1''</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c13">a1''</font>, <font color="Maroon" title="c14">b1''</font> being    <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="c2">o2</font><b> such that </b><br/><a NAME="E17:20_2"/><i><font color="Green" title="E34">A29</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">a1''</font>,<font color="Maroon" title="c14">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E18:20_2"/><i><font color="Green" title="E35">A30</font></i>: 
<font color="Maroon" title="c10">Gi</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c13">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c14">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E3:20_2"><i><font color="Green" title="E25">A20</font></i></a>, <a class="txt" href="#E6:20_2"><i><font color="Green" title="E28">A23</font></i></a>, <a class="txt" href="#E8:20_2"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="#E14:20_2"><i><font color="Green" title="E32">A27</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">a1''</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1''</font> &amp; <font color="Maroon" title="c11">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c13">a1''</font> )</span><br/>

<a NAME="E19:20_2"/>
<font color="Maroon" title="c12">m</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="c4">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:20_2"><i><font color="Green" title="E25">A20</font></i></a>, <a class="txt" href="#E5:20_2"><i><font color="Green" title="E27">A22</font></i></a>, <a class="txt" href="#E8:20_2"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="#E14:20_2"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="#E15:20_2"><i><font color="Green" title="E33">A28</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>hence </b><a NAME="E20:20_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1''</font>
 <b>by </b><i><a class="txt" href="#E17:20_2"><i><font color="Green" title="E34">A29</font></i></a>, <a class="ref" href="polynom1.html#T74" title="POLYNOM1:th.74">POLYNOM1:74</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c13">a1''</font></span><br/>

<b>consider </b><font color="Maroon" title="c15">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E22:20_2"/><i><font color="Green" title="E36">A31</font></i>: 
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font>
 <b>and </b><br/><a NAME="E23:20_2"/><i><font color="Green" title="E37">A32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="#E6:20"><i><font color="Green" title="E20">A15</font></i></a>, <a class="txt" href="#E7:20_2"><i><font color="Green" title="E29">A24</font></i></a>, <a class="txt" href="#E15:20_2"><i><font color="Green" title="E33">A28</font></i></a></i>;<br/>
<a NAME="E24:20_2"/><i><font color="Green" title="E38">A33</font></i>: 
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c13">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c14">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E3:20_2"><i><font color="Green" title="E25">A20</font></i></a>, <a class="txt" href="#E8:20_2"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="#E14:20_2"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="#E15:20_2"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="#E18:20_2"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="#E22:20_2"><i><font color="Green" title="E36">A31</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c11">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> = 
<font color="Maroon" title="c11">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font>
<b>by </b><i><a class="txt" href="#E15:20_2"><i><font color="Green" title="E33">A28</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c8">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c13">a1''</font>
<b>by </b><i><a class="txt" href="#E23:20_2"><i><font color="Green" title="E37">A32</font></i></a>, <a class="txt" href="#E24:20_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</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="E11:20"/><b>then </b><i><font color="Green" title="E24">A34</font></i>: 
 <a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c6">F</font>
 
<b>by </b><i><a class="txt" href="#E8:20"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="polynom6.html#T12" target="_self" title="POLYNOM6:th.12">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">FG</font> =  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">bb</font> = <font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> ;<br/>
<a NAME="E14:20"/>
 <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="c4">b2</font></span>)</span> <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#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E15:20"/><b>then </b><i><font color="Green" title="E25">A35</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E16:20"/><i><font color="Green" title="E26">A36</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> <font color="Maroon" title="c3">b1</font></span>)</span> <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#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/>
<i><font color="Green" title="E27">A37</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#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
<b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font>
<b>by </b><i><a class="txt" href="#E5:20"><i><font color="Green" title="E19">A14</font></i></a>, <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#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c5">G</font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>
;<br/>
<a NAME="E18:20"/><b>then </b><i><font color="Green" title="E28">A38</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c5">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E19:20"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">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#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> implies <span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> )</span><br/>

<b>assume </b><a NAME="E1:20_4"/><i><font color="Green" title="E29">A39</font></i>: 
<font color="Maroon" title="c9">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#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span><br/>

<a NAME="E2:20_4"/><b>then </b><i><font color="Green" title="E30">A40</font></i>: 
<font color="Maroon" title="c9">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#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span>
 
;<br/>
<a NAME="E3:20_4"/><i><font color="Green" title="E31">A41</font></i>: 
<font color="Maroon" title="c9">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#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c5">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E17:20"><i><font color="Green" title="E27">A37</font></i></a>, <a class="txt" href="#E1:20_4"><i><font color="Green" title="E29">A39</font></i></a></i>;<br/>
<a NAME="E4:20_4"/><i><font color="Green" title="E32">A42</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#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>
<a NAME="E5:20_4"/><i><font color="Green" title="E33">A43</font></i>: 
<font color="Maroon" title="c9">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> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="#E3:20_4"><i><font color="Green" title="E31">A41</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>
<a NAME="E6:20_4"/><i><font color="Green" title="E34">A44</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:20_4"><i><font color="Green" title="E30">A40</font></i></a>, <a class="txt" href="#E4:20_4"><i><font color="Green" title="E32">A42</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>
<a NAME="E7:20_4"/><i><font color="Green" title="E35">A45</font></i>: 
<font color="Maroon" title="c9">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> <font color="Maroon" title="c3">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E5:20"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="#E2:20_4"><i><font color="Green" title="E30">A40</font></i></a>, <a class="txt" href="#E4:20_4"><i><font color="Green" title="E32">A42</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">a1'</font> being    <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">o1</font>, <font color="Maroon" title="c11">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span><b> such that </b><br/><a NAME="E9:20_4"/><i><font color="Green" title="E36">A46</font></i>: 
<font color="Maroon" title="c11">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c9">j</font>
 <b>and </b><br/><a NAME="E10:20_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a1'</font>
 <b>and </b><br/><a NAME="E11:20_4"/><i><font color="Green" title="E37">A47</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:20_4"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</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="c11">Fk</font> holds <br/> ex <font color="Olive" title="b2">a1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1''</font> &amp; <font color="Maroon" title="c11">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a1''</font> )
 <b>by </b><i><a class="txt" href="#E9:20"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="#E10:20"><i><font color="Green" title="E23">A18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">a2'</font>, <font color="Maroon" title="c13">b2'</font> being    <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">o1</font>, <font color="Maroon" title="c14">Gk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E14:20_4"/><i><font color="Green" title="E38">A48</font></i>: 
<font color="Maroon" title="c14">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c9">j</font>
 <b>and </b><br/><a NAME="E15:20_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c12">a2'</font>,<font color="Maroon" title="c13">b2'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E16:20_4"/><i><font color="Green" title="E39">A49</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E17:20_4"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</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="c14">Gk</font> holds <br/> ex <font color="Olive" title="b2">a2''</font>, <font color="Olive" title="b3">b2''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">a2''</font>,<font color="Olive" title="b3">b2''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c14">Gk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a2'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a2''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b2'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b2''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="#E2:20"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="#E7:20_4"><i><font color="Green" title="E35">A45</font></i></a></i>;<br/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c9">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:20_4"><i><font color="Green" title="E30">A40</font></i></a>, <a class="txt" href="#E4:20_4"><i><font color="Green" title="E32">A42</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E15:20"><i><font color="Green" title="E25">A35</font></i></a>, <a class="txt" href="#E5:20_4"><i><font color="Green" title="E33">A43</font></i></a>, <a class="txt" href="#E9:20_4"><i><font color="Green" title="E36">A46</font></i></a>, <a class="txt" href="#E11:20_4"><i><font color="Green" title="E37">A47</font></i></a>, <a class="txt" href="#E14:20_4"><i><font color="Green" title="E38">A48</font></i></a>, <a class="txt" href="#E16:20_4"><i><font color="Green" title="E39">A49</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E19:20_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="#E5:20_4"><i><font color="Green" title="E33">A43</font></i></a>, <a class="txt" href="#E6:20_4"><i><font color="Green" title="E34">A44</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</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="E20:20"/><b>then </b><i><font color="Green" title="E29">A50</font></i>: 
 <a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="#E18:20"><i><font color="Green" title="E28">A38</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E21:20"/><b>then </b><i><font color="Green" title="E30">A51</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c6">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T31" title="POLYNOM1:th.31">POLYNOM1:31</a></i>;<br/>
<a NAME="E22:20"/><b>then </b><i><font color="Green" title="E31">A52</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">FG</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#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c8">bb</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E11:20"><i><font color="Green" title="E24">A34</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E23:20"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">p</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font>  st <font color="Olive" title="b1">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="c7">FG</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c8">bb</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c7">FG</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">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="Olive" title="b2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">bb</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Olive" title="b2">p</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font>  st <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> <font color="Maroon" title="c7">FG</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c8">bb</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> holds <br/><font color="Maroon" title="c7">FG</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c9">k</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="Olive" title="b1">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">bb</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Olive" title="b1">p</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><br/><b>let </b><font color="Maroon" title="c10">p</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> <font color="Maroon" title="c7">FG</font> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c8">bb</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> implies <font color="Maroon" title="c7">FG</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c9">k</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="c10">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">bb</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c10">p</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:20_5"/><i><font color="Green" title="E32">A53</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> <font color="Maroon" title="c7">FG</font>
 <b>and </b><br/><a NAME="E2:20_5"/><i><font color="Green" title="E33">A54</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c8">bb</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">FG</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c9">k</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="c10">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">bb</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c10">p</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><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="E4:20_5"/><i><font color="Green" title="E34">A55</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="c5">G</font>
 <b>and </b><br/><a NAME="E5:20_5"/><i><font color="Green" title="E35">A56</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="c5">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="E6:20_5"/><i><font color="Green" title="E36">A57</font></i>: 
<font color="Maroon" title="c9">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="c5">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="E7:20_5"/><i><font color="Green" title="E37">A58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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> <font color="Maroon" title="c7">FG</font> <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="txt" href="#E1:20_5"><i><font color="Green" title="E32">A53</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/>
<a NAME="E8:20_5"/>
<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="c6">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E21:20"><i><font color="Green" title="E30">A51</font></i></a>, <a class="txt" href="#E1:20_5"><i><font color="Green" title="E32">A53</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">i'</font>, <font color="Maroon" title="c14">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="E10:20_5"/><i><font color="Green" title="E38">A59</font></i>: 
<font color="Maroon" title="c13">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="c6">F</font>
 <b>and </b><br/><a NAME="E11:20_5"/><i><font color="Green" title="E39">A60</font></i>: 
<font color="Maroon" title="c14">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="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i'</font></span>)</span>
 <b>and </b><br/><a NAME="E12:20_5"/><i><font color="Green" title="E40">A61</font></i>: 
<font color="Maroon" title="c9">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="c6">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c14">j'</font>
 <b>and </b><br/><a NAME="E13:20_5"/><i><font color="Green" title="E41">A62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c6">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="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/>
<a NAME="E14:20_5"/>
(  <a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<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> &amp;  <a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i'</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K4" title="POLYNOM1:func.4">Card</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i'</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="polynom3.html#T16" title="POLYNOM3:th.16">POLYNOM3:16</a></i>;<br/>
<a NAME="E15:20_5"/><b>then </b><i><font color="Green" title="E42">A63</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i'</font> &amp; <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j'</font> )
 
<b>by </b><i><a class="txt" href="#E20:20"><i><font color="Green" title="E29">A50</font></i></a>, <a class="txt" href="#E4:20_5"><i><font color="Green" title="E34">A55</font></i></a>, <a class="txt" href="#E5:20_5"><i><font color="Green" title="E35">A56</font></i></a>, <a class="txt" href="#E6:20_5"><i><font color="Green" title="E36">A57</font></i></a>, <a class="txt" href="#E10:20_5"><i><font color="Green" title="E38">A59</font></i></a>, <a class="txt" href="#E11:20_5"><i><font color="Green" title="E39">A60</font></i></a>, <a class="txt" href="#E12:20_5"><i><font color="Green" title="E40">A61</font></i></a>, <a class="ref" href="polynom3.html#T22" title="POLYNOM3:th.22">POLYNOM3:22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">a1'</font>, <font color="Maroon" title="c16">b1'</font> being    <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">o1</font>, <font color="Maroon" title="c17">Gi</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E17:20_5"/><i><font color="Green" title="E43">A64</font></i>: 
<font color="Maroon" title="c17">Gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c11">i</font>
 <b>and </b><br/><a NAME="E18:20_5"/><i><font color="Green" title="E44">A65</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c15">a1'</font>,<font color="Maroon" title="c16">b1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E19:20_5"/><i><font color="Green" title="E45">A66</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">Gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E20:20_5"/><i><font color="Green" title="E46">A67</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</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="c17">Gi</font> holds <br/> ex <font color="Olive" title="b2">a1''</font>, <font color="Olive" title="b3">b1''</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">a1''</font>,<font color="Olive" title="b3">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c17">Gi</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E2:20"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="#E4:20_5"><i><font color="Green" title="E34">A55</font></i></a></i>;<br/>
<a NAME="E21:20_5"/><i><font color="Green" title="E47">A68</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> <font color="Maroon" title="c17">Gi</font>
 
<b>by </b><i><a class="txt" href="#E4:20_5"><i><font color="Green" title="E34">A55</font></i></a>, <a class="txt" href="#E5:20_5"><i><font color="Green" title="E35">A56</font></i></a>, <a class="txt" href="#E17:20_5"><i><font color="Green" title="E43">A64</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">a1''</font>, <font color="Maroon" title="c19">b1''</font> being    <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="c2">o2</font><b> such that </b><br/><a NAME="E23:20_5"/><i><font color="Green" title="E48">A69</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN: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="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c18">a1''</font>,<font color="Maroon" title="c19">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E24:20_5"/><i><font color="Green" title="E49">A70</font></i>: 
<font color="Maroon" title="c17">Gi</font> <a href="matrlin.html#K1" title="MATRLIN: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="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c18">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E20:20_5"><i><font color="Green" title="E46">A67</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">fa1</font> being    <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">o1</font>, <font color="Maroon" title="c21">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span><b> such that </b><br/><a NAME="E26:20_5"/><i><font color="Green" title="E50">A71</font></i>: 
<font color="Maroon" title="c21">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c11">i</font>
 <b>and </b><br/><a NAME="E27:20_5"/><i><font color="Green" title="E51">A72</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">fa1</font>
 <b>and </b><br/><a NAME="E28:20_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E29:20_5"/><i><font color="Green" title="E52">A73</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</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="c21">Fk</font> holds <br/> ex <font color="Olive" title="b2">fa1'</font> being   <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="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">fa1'</font> &amp; <font color="Maroon" title="c21">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">fa1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">fa1'</font> )
 <b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E10:20"><i><font color="Green" title="E23">A18</font></i></a>, <a class="txt" href="#E16:20"><i><font color="Green" title="E26">A36</font></i></a>, <a class="txt" href="#E4:20_5"><i><font color="Green" title="E34">A55</font></i></a></i>;<br/>
<a NAME="E30:20_5"/><i><font color="Green" title="E53">A74</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> <font color="Maroon" title="c21">Fk</font>
 
<b>by </b><i><a class="txt" href="#E10:20_5"><i><font color="Green" title="E38">A59</font></i></a>, <a class="txt" href="#E11:20_5"><i><font color="Green" title="E39">A60</font></i></a>, <a class="txt" href="#E15:20_5"><i><font color="Green" title="E42">A63</font></i></a>, <a class="txt" href="#E26:20_5"><i><font color="Green" title="E50">A71</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">fa1'</font> being    <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="c2">o2</font><b> such that </b><br/><a NAME="E32:20_5"/><i><font color="Green" title="E54">A75</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">fa1'</font>
 <b>and </b><br/><a NAME="E33:20_5"/><i><font color="Green" title="E55">A76</font></i>: 
<font color="Maroon" title="c21">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">fa1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c22">fa1'</font>
 <b>by </b><i><a class="txt" href="#E29:20_5"><i><font color="Green" title="E52">A73</font></i></a></i>;<br/>
<a NAME="E34:20_5"/><i><font color="Green" title="E56">A77</font></i>: 
<font color="Maroon" title="c15">a1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c20">fa1</font>
 
<b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E4:20_5"><i><font color="Green" title="E34">A55</font></i></a>, <a class="txt" href="#E18:20_5"><i><font color="Green" title="E44">A65</font></i></a>, <a class="txt" href="#E27:20_5"><i><font color="Green" title="E51">A72</font></i></a>, <a class="ref" href="polynom1.html#T74" title="POLYNOM1:th.74">POLYNOM1:74</a></i>;<br/>
<a NAME="E35:20_5"/><i><font color="Green" title="E57">A78</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"><a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E19:20_5"><i><font color="Green" title="E45">A66</font></i></a>, <a class="txt" href="#E21:20_5"><i><font color="Green" title="E47">A68</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E36:20_5"/><b>then </b><i><font color="Green" title="E58">A79</font></i>: 
<font color="Maroon" title="c18">a1''</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">fa1'</font>
 
<b>by </b><i><a class="txt" href="#E23:20_5"><i><font color="Green" title="E48">A69</font></i></a>, <a class="txt" href="#E32:20_5"><i><font color="Green" title="E54">A75</font></i></a>, <a class="ref" href="polynom1.html#T74" title="POLYNOM1:th.74">POLYNOM1:74</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">b1a1</font> = <font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c15">a1'</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">o1</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">b2a1</font> = <font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c18">a1''</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="c2">o2</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E39:20_5"/>
<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="c6">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E21:20"><i><font color="Green" title="E30">A51</font></i></a>, <a class="txt" href="#E1:20_5"><i><font color="Green" title="E32">A53</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>then </b><i><font color="Green" title="E59">A80</font></i>: <font color="Maroon" title="c10">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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>
<b>by </b><i><a class="txt" href="#E11:20"><i><font color="Green" title="E24">A34</font></i></a>, <a class="txt" href="#E2:20_5"><i><font color="Green" title="E33">A54</font></i></a>, <a class="txt" href="#E13:20_5"><i><font color="Green" title="E41">A62</font></i></a>, <a class="txt" href="#E15:20_5"><i><font color="Green" title="E42">A63</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c21">Fk</font> <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="#E10:20_5"><i><font color="Green" title="E38">A59</font></i></a>, <a class="txt" href="#E15:20_5"><i><font color="Green" title="E42">A63</font></i></a>, <a class="txt" href="#E26:20_5"><i><font color="Green" title="E50">A71</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c15">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c18">a1''</font>
<b>by </b><i><a class="txt" href="#E30:20_5"><i><font color="Green" title="E53">A74</font></i></a>, <a class="txt" href="#E33:20_5"><i><font color="Green" title="E55">A76</font></i></a>, <a class="txt" href="#E34:20_5"><i><font color="Green" title="E56">A77</font></i></a>, <a class="txt" href="#E36:20_5"><i><font color="Green" title="E58">A79</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<a NAME="E41:20_5"/><i><font color="Green" title="E60">A81</font></i>: 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c15">a1'</font>,<font color="Maroon" title="c16">b1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <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">a1'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c15">a1'</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E4:20_5"><i><font color="Green" title="E34">A55</font></i></a>, <a class="txt" href="#E18:20_5"><i><font color="Green" title="E44">A65</font></i></a>, <a class="txt" href="#E27:20_5"><i><font color="Green" title="E51">A72</font></i></a>, <a class="txt" href="#E34:20_5"><i><font color="Green" title="E56">A77</font></i></a>, <a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E42:20_5"/><i><font color="Green" title="E61">A82</font></i>: 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c18">a1''</font>,<font color="Maroon" title="c19">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <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">a1''</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c18">a1''</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E23:20_5"><i><font color="Green" title="E48">A69</font></i></a>, <a class="txt" href="#E32:20_5"><i><font color="Green" title="E54">A75</font></i></a>, <a class="txt" href="#E35:20_5"><i><font color="Green" title="E57">A78</font></i></a>, <a class="txt" href="#E36:20_5"><i><font color="Green" title="E58">A79</font></i></a>, <a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/>
<i><font color="Green" title="E62">A83</font></i>: <font color="Maroon" title="c8">bb</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c10">p</font> = 
<font color="Maroon" title="c23">b1a1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c24">b2a1</font>
<b>by </b><i><a class="txt" href="#E40:20_5"><i><font color="Green" title="E59">A80</font></i></a>, <a class="ref" href="polynom6.html#T13" target="_self" title="POLYNOM6:th.13">Th13</a></i>
<br/>.= 
<font color="Maroon" title="c16">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c24">b2a1</font>
<b>by </b><i><a class="txt" href="#E41:20_5"><i><font color="Green" title="E60">A81</font></i></a>, <a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>
<br/>.= 
<font color="Maroon" title="c16">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b1''</font>
<b>by </b><i><a class="txt" href="#E42:20_5"><i><font color="Green" title="E61">A82</font></i></a>, <a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>
;<br/>
<b>thus </b><font color="Maroon" title="c7">FG</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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>
<b>by </b><i><a class="txt" href="#E1:20_5"><i><font color="Green" title="E32">A53</font></i></a>, <a class="txt" href="#E7:20_5"><i><font color="Green" title="E37">A58</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c17">Gi</font> <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="#E4:20_5"><i><font color="Green" title="E34">A55</font></i></a>, <a class="txt" href="#E17:20_5"><i><font color="Green" title="E43">A64</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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="c10">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">bb</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c10">p</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E21:20_5"><i><font color="Green" title="E47">A68</font></i></a>, <a class="txt" href="#E24:20_5"><i><font color="Green" title="E49">A70</font></i></a>, <a class="txt" href="#E40:20_5"><i><font color="Green" title="E59">A80</font></i></a>, <a class="txt" href="#E43:20_5"><i><font color="Green" title="E62">A83</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1: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>
<b>hence </b><a NAME="E24:20"/>
 <a href="polynom1.html#K19" title="POLYNOM1:func.19">decomp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="#E22:20"><i><font color="Green" title="E31">A52</font></i></a>, <a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</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>
