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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#K13" title="POLYNOM1:func.13">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#K13" title="POLYNOM1:func.13">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"><a href="polynom1.html#K13" title="POLYNOM1:func.13">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  <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">divisors</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">divisors</a> <font color="Olive" title="b1">b1</font></span>)</span> 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#K13" title="POLYNOM1:func.13">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  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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="b6">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">divisors</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a1'</font> &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">divisors</a> <font color="Olive" title="b2">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> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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">divisors</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">a1''</font> &amp; <font color="Olive" title="b6">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b8">a1''</font> ) ) ) ) holds <br/> <a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</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#K13" title="POLYNOM1:func.13">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#K13" title="POLYNOM1:func.13">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"><a href="polynom1.html#K13" title="POLYNOM1:func.13">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  <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">divisors</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">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> 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#K13" title="POLYNOM1:func.13">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  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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="b5">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">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="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a1'</font> &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">divisors</a> <font color="Olive" title="b1">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> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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">divisors</a> <font color="Olive" title="b1">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">a1''</font> &amp; <font color="Olive" title="b5">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b7">a1''</font> ) ) ) ) holds <br/> <a href="polynom1.html#K19" title="POLYNOM1:func.19">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="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#K13" title="POLYNOM1:func.13">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"><a href="polynom1.html#K13" title="POLYNOM1:func.13">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  <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">divisors</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">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> 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#K13" title="POLYNOM1:func.13">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  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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="b4">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">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="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a1'</font> &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">divisors</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> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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">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="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">a1''</font> &amp; <font color="Olive" title="b4">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b6">a1''</font> ) ) ) ) holds <br/> <a href="polynom1.html#K19" title="POLYNOM1:func.19">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="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"><a href="polynom1.html#K13" title="POLYNOM1:func.13">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> ; <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">divisors</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">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#K13" title="POLYNOM1:func.13">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#K13" title="POLYNOM1:func.13">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="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">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#K19" title="POLYNOM1:func.19">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#K13" title="POLYNOM1:func.13">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">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> ) ) ) ) implies  <a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c5">G</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green" title="E14">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">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green" title="E15">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">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#K13" title="POLYNOM1:func.13">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#K13" title="POLYNOM1:func.13">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="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">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#K19" title="POLYNOM1:func.19">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#K13" title="POLYNOM1:func.13">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">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> ) ) )
 ; <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">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="c5">G</font></span><br/>

<a NAME="E3:18"/>
 <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">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E4:18"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:18"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
not <font color="Maroon" title="c5">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><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> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="polynom3.html#K1" title="POLYNOM3:func.1">Del</a> <font color="Maroon" title="c5">G</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="polyalg1.html#T4" title="POLYALG1:th.4">POLYALG1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">D</font> =  <a href="polynom3.html#K1" title="POLYNOM3:func.1">Del</a> <font color="Maroon" title="c5">G</font>,1 as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">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>  ;<br/>
<a NAME="E8:18"/><i><font color="Green" title="E18">A5</font></i>: 
1 <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="polynom6.html#E5:18"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">a1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c8">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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="E10:18"/><i><font color="Green" title="E19">A6</font></i>: 
( <font color="Maroon" title="c8">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> 1 &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1'</font> )
 <b>and </b><br/><a NAME="E11:18"/><i><font color="Green" title="E20">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:18"/>
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="c8">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#K13" title="POLYNOM1:func.13">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">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="c8">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="c7">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="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">G1</font> = <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 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#K13" title="POLYNOM1:func.13">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="txt" href="polynom6.html#E8:18"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="polynom6.html#E10:18"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<i><font color="Green" title="E21">A8</font></i>:  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font> = 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <span class="p2"><a href="matrlin.html#K3" title="MATRLIN:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c9">G1</font></span><a href="matrlin.html#K3" title="MATRLIN:func.3">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c6">D</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E6:18"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="dtconstr.html#T21" title="DTCONSTR:th.21">DTCONSTR:21</a></i>
<br/>.= 
<font color="Maroon" title="c9">G1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c6">D</font></span>)</span>
<b>by </b><i><a class="ref" href="dtconstr.html#T13" title="DTCONSTR:th.13">DTCONSTR:13</a></i>
;<br/>
<a NAME="E15:18"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c8">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="polynom6.html#E8:18"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="polynom6.html#E10:18"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E16:18"/>
 <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">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E17:18"/><b>then </b>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:18"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="polynom6.html#E15:18"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<a NAME="E18:18"/><b>then </b>
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="polynom6.html#E14:18"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="finseq_1.html#T48" title="FINSEQ_1:th.48">FINSEQ_1:48</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">S</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">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>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">F</font> =  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font>;<br/>
<a NAME="E20:18"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/>
<a NAME="E21:18"/><b>then </b><i><font color="Green" title="E23">A10</font></i>: 
 <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</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> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c10">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T133" title="ORDERS_1:th.133">ORDERS_1:133</a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/>
<a NAME="E22:18"/>
for <font color="Olive" title="b1">n</font>, <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="b1">n</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="c5">G</font></span>)</span> &amp; <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> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/>( <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">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">n</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="c5">G</font></span>)</span> &amp; <font color="Maroon" title="c13">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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> implies ( <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</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> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/><i><font color="Green" title="E24">A11</font></i>: 
( <font color="Maroon" title="c12">n</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="c5">G</font></span>)</span> &amp; <font color="Maroon" title="c13">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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green" title="E25">A12</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</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> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green" title="E26">A13</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</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="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:18_2_1"/><i><font color="Green" title="E27">A14</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</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="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E3:18_2_1"/><i><font color="Green" title="E28">A15</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</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="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">i1</font>, <font color="Maroon" title="c15">j1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:18_2_1"/><i><font color="Green" title="E29">A16</font></i>: 
<font color="Maroon" title="c14">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font>
 <b>and </b><br/><a NAME="E6:18_2_1"/><i><font color="Green" title="E30">A17</font></i>: 
<font color="Maroon" title="c15">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i1</font></span>)</span>
 <b>and </b><br/><a NAME="E7:18_2_1"/><i><font color="Green" title="E31">A18</font></i>: 
<font color="Maroon" title="c12">n</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#K5" title="POLYNOM1:func.5">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="c14">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c15">j1</font>
 <b>and </b><br/><a NAME="E8:18_2_1"/><i><font color="Green" title="E32">A19</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="c14">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">i2</font>, <font color="Maroon" title="c17">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:18_2_1"/><i><font color="Green" title="E33">A20</font></i>: 
<font color="Maroon" title="c16">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font>
 <b>and </b><br/><a NAME="E11:18_2_1"/><i><font color="Green" title="E34">A21</font></i>: 
<font color="Maroon" title="c17">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:18_2_1"/><i><font color="Green" title="E35">A22</font></i>: 
<font color="Maroon" title="c13">m</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#K5" title="POLYNOM1:func.5">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="c16">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c17">j2</font>
 <b>and </b><br/><a NAME="E13:18_2_1"/><i><font color="Green" title="E36">A23</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="c16">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">a1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c19">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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="E15:18_2_1"/><i><font color="Green" title="E37">A24</font></i>: 
<font color="Maroon" title="c19">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="Maroon" title="c14">i1</font>
 <b>and </b><br/><a NAME="E16:18_2_1"/><i><font color="Green" title="E38">A25</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a1'</font>
 <b>and </b><br/><a NAME="E17:18_2_1"/><i><font color="Green" title="E39">A26</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">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">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E18:18_2_1"/><i><font color="Green" title="E40">A27</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">r</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="c19">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#K13" title="POLYNOM1:func.13">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">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">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1''</font> &amp; <font color="Maroon" title="c19">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">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="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="polynom6.html#E5:18_2_1"><i><font color="Green" title="E29">A16</font></i></a></i>;<br/><a NAME="E19:18_2_1"/><i><font color="Green" title="E41">A28</font></i>: 
<font color="Maroon" title="c15">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">Fk</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:18_2_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="polynom6.html#E6:18_2_1"><i><font color="Green" title="E30">A17</font></i></a>, <a class="txt" href="polynom6.html#E15:18_2_1"><i><font color="Green" title="E37">A24</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="c20">a1''</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E21:18_2_1"/><i><font color="Green" title="E42">A29</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1''</font>
 <b>and </b><br/><a NAME="E22:18_2_1"/><i><font color="Green" title="E43">A30</font></i>: 
<font color="Maroon" title="c19">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c20">a1''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E18:18_2_1"><i><font color="Green" title="E40">A27</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c21">a2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c22">Fk'</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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="E24:18_2_1"/><i><font color="Green" title="E44">A31</font></i>: 
<font color="Maroon" title="c22">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="Maroon" title="c16">i2</font>
 <b>and </b><br/><a NAME="E25:18_2_1"/><i><font color="Green" title="E45">A32</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c16">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">a2'</font>
 <b>and </b><br/><a NAME="E26:18_2_1"/><i><font color="Green" title="E46">A33</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">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">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E27:18_2_1"/><i><font color="Green" title="E47">A34</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">r</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="c22">Fk'</font> holds <br/> ex <font color="Olive" title="b2">a2''</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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">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">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2''</font> &amp; <font color="Maroon" title="c22">Fk'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">a2'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a2''</font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="polynom6.html#E10:18_2_1"><i><font color="Green" title="E33">A20</font></i></a></i>;<br/><a NAME="E28:18_2_1"/><i><font color="Green" title="E48">A35</font></i>: 
<font color="Maroon" title="c17">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">Fk'</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E10:18_2_1"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="polynom6.html#E11:18_2_1"><i><font color="Green" title="E34">A21</font></i></a>, <a class="txt" href="polynom6.html#E24:18_2_1"><i><font color="Green" title="E44">A31</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="c23">a2''</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E30:18_2_1"/><i><font color="Green" title="E49">A36</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c17">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a2''</font>
 <b>and </b><br/><a NAME="E31:18_2_1"/><i><font color="Green" title="E50">A37</font></i>: 
<font color="Maroon" title="c22">Fk'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">a2'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c23">a2''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E27:18_2_1"><i><font color="Green" title="E47">A34</font></i></a></i>;<br/><a NAME="E32:18_2_1"/>
<font color="Maroon" title="c19">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:18_2_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="polynom6.html#E15:18_2_1"><i><font color="Green" title="E37">A24</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E33:18_2_1"/><b>then </b><i><font color="Green" title="E51">A38</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</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="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E6:18_2_1"><i><font color="Green" title="E30">A17</font></i></a>, <a class="txt" href="polynom6.html#E8:18_2_1"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E34:18_2_1"/>
<font color="Maroon" title="c22">Fk'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E10:18_2_1"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="polynom6.html#E24:18_2_1"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E35:18_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</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="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">Fk'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E11:18_2_1"><i><font color="Green" title="E34">A21</font></i></a>, <a class="txt" href="polynom6.html#E13:18_2_1"><i><font color="Green" title="E36">A23</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E36:18_2_1"/><b>then </b><i><font color="Green" title="E52">A39</font></i>: 
( <font color="Maroon" title="c18">a1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c21">a2'</font> &amp; <font color="Maroon" title="c20">a1''</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">a2''</font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18_2_1"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="polynom6.html#E2:18_2_1"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="polynom6.html#E3:18_2_1"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="polynom6.html#E22:18_2_1"><i><font color="Green" title="E43">A30</font></i></a>, <a class="txt" href="polynom6.html#E31:18_2_1"><i><font color="Green" title="E50">A37</font></i></a>, <a class="txt" href="polynom6.html#E33:18_2_1"><i><font color="Green" title="E51">A38</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/><a NAME="E37:18_2_1"/><i><font color="Green" title="E53">A40</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a1'</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E5:18_2_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="polynom6.html#E16:18_2_1"><i><font color="Green" title="E38">A25</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E38:18_2_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">a2'</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E10:18_2_1"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="polynom6.html#E25:18_2_1"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E39:18_2_1"/><b>then </b><i><font color="Green" title="E54">A41</font></i>: 
<font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E5:18_2_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="polynom6.html#E10:18_2_1"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="polynom6.html#E36:18_2_1"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="polynom6.html#E37:18_2_1"><i><font color="Green" title="E53">A40</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E40:18_2_1"/>
 <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="c19">Fk</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="c19">Fk</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E41:18_2_1"/><b>then </b><i><font color="Green" title="E55">A42</font></i>: 
<font color="Maroon" title="c15">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E17:18_2_1"><i><font color="Green" title="E39">A26</font></i></a>, <a class="txt" href="polynom6.html#E19:18_2_1"><i><font color="Green" title="E41">A28</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="E42:18_2_1"/><b>then </b><i><font color="Green" title="E56">A43</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E21:18_2_1"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E43:18_2_1"/>
 <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="c22">Fk'</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="c22">Fk'</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E44:18_2_1"/><b>then </b><i><font color="Green" title="E57">A44</font></i>: 
<font color="Maroon" title="c17">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E26:18_2_1"><i><font color="Green" title="E46">A33</font></i></a>, <a class="txt" href="polynom6.html#E28:18_2_1"><i><font color="Green" title="E48">A35</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="E45:18_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a2''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E30:18_2_1"><i><font color="Green" title="E49">A36</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="E46:18_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom6.html#E2:18_2"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="polynom6.html#E7:18_2_1"><i><font color="Green" title="E31">A18</font></i></a>, <a class="txt" href="polynom6.html#E12:18_2_1"><i><font color="Green" title="E35">A22</font></i></a>, <a class="txt" href="polynom6.html#E36:18_2_1"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="polynom6.html#E39:18_2_1"><i><font color="Green" title="E54">A41</font></i></a>, <a class="txt" href="polynom6.html#E41:18_2_1"><i><font color="Green" title="E55">A42</font></i></a>, <a class="txt" href="polynom6.html#E42:18_2_1"><i><font color="Green" title="E56">A43</font></i></a>, <a class="txt" href="polynom6.html#E44:18_2_1"><i><font color="Green" title="E57">A44</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2"/>
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</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></span><br/>

<a NAME="E5:18_2"/><i><font color="Green" title="E26">A45</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</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="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:18_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E6:18_2"/><i><font color="Green" title="E27">A46</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</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="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:18_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">Fn</font> = <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c15">Fm</font> = <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</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> <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="c16">Fn'</font> = <font color="Maroon" title="c14">Fn</font>, <font color="Maroon" title="c17">Fm'</font> = <font color="Maroon" title="c15">Fm</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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>consider </b><font color="Maroon" title="c18">a1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c19">a2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E10:18_2"/><i><font color="Green" title="E28">A47</font></i>: 
<font color="Maroon" title="c16">Fn'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">a1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">a2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">c1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c21">c2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E12:18_2"/><i><font color="Green" title="E29">A48</font></i>: 
<font color="Maroon" title="c17">Fm'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c20">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">c2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">i1</font>, <font color="Maroon" title="c23">j1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E14:18_2"/><i><font color="Green" title="E30">A49</font></i>: 
<font color="Maroon" title="c22">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font>
 <b>and </b><br/><a NAME="E15:18_2"/><i><font color="Green" title="E31">A50</font></i>: 
<font color="Maroon" title="c23">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span>
 <b>and </b><br/><a NAME="E16:18_2"/><i><font color="Green" title="E32">A51</font></i>: 
<font color="Maroon" title="c12">n</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#K5" title="POLYNOM1:func.5">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="c22">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c23">j1</font>
 <b>and </b><br/><a NAME="E17:18_2"/><i><font color="Green" title="E33">A52</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="c22">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">i2</font>, <font color="Maroon" title="c25">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E19:18_2"/><i><font color="Green" title="E34">A53</font></i>: 
<font color="Maroon" title="c24">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font>
 <b>and </b><br/><a NAME="E20:18_2"/><i><font color="Green" title="E35">A54</font></i>: 
<font color="Maroon" title="c25">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i2</font></span>)</span>
 <b>and </b><br/><a NAME="E21:18_2"/><i><font color="Green" title="E36">A55</font></i>: 
<font color="Maroon" title="c13">m</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#K5" title="POLYNOM1:func.5">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="c24">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">j2</font>
 <b>and </b><br/><a NAME="E22:18_2"/><i><font color="Green" title="E37">A56</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="c24">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">a1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c27">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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="E24:18_2"/><i><font color="Green" title="E38">A57</font></i>: 
<font color="Maroon" title="c27">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="Maroon" title="c22">i1</font>
 <b>and </b><br/><a NAME="E25:18_2"/><i><font color="Green" title="E39">A58</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a1'</font>
 <b>and </b><br/><a NAME="E26:18_2"/><i><font color="Green" title="E40">A59</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">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">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E27:18_2"/><i><font color="Green" title="E41">A60</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">r</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="c27">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#K13" title="POLYNOM1:func.13">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">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">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1''</font> &amp; <font color="Maroon" title="c27">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">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="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="polynom6.html#E14:18_2"><i><font color="Green" title="E30">A49</font></i></a></i>;<br/>
<a NAME="E28:18_2"/><i><font color="Green" title="E42">A61</font></i>: 
<font color="Maroon" title="c23">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">Fk</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E14:18_2"><i><font color="Green" title="E30">A49</font></i></a>, <a class="txt" href="polynom6.html#E15:18_2"><i><font color="Green" title="E31">A50</font></i></a>, <a class="txt" href="polynom6.html#E24:18_2"><i><font color="Green" title="E38">A57</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="c28">a1''</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E30:18_2"/><i><font color="Green" title="E43">A62</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c23">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a1''</font>
 <b>and </b><br/><a NAME="E31:18_2"/><i><font color="Green" title="E44">A63</font></i>: 
<font color="Maroon" title="c27">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c28">a1''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E27:18_2"><i><font color="Green" title="E41">A60</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">a2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c30">Fk'</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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="E33:18_2"/><i><font color="Green" title="E45">A64</font></i>: 
<font color="Maroon" title="c30">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="Maroon" title="c24">i2</font>
 <b>and </b><br/><a NAME="E34:18_2"/><i><font color="Green" title="E46">A65</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c24">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">a2'</font>
 <b>and </b><br/><a NAME="E35:18_2"/><i><font color="Green" title="E47">A66</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">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">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E36:18_2"/><i><font color="Green" title="E48">A67</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">r</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="c30">Fk'</font> holds <br/> ex <font color="Olive" title="b2">a2''</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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">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">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2''</font> &amp; <font color="Maroon" title="c30">Fk'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">a2'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a2''</font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="polynom6.html#E19:18_2"><i><font color="Green" title="E34">A53</font></i></a></i>;<br/>
<a NAME="E37:18_2"/><i><font color="Green" title="E49">A68</font></i>: 
<font color="Maroon" title="c25">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c30">Fk'</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E19:18_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="txt" href="polynom6.html#E20:18_2"><i><font color="Green" title="E35">A54</font></i></a>, <a class="txt" href="polynom6.html#E33:18_2"><i><font color="Green" title="E45">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="c31">a2''</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E39:18_2"/><i><font color="Green" title="E50">A69</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c25">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">a2''</font>
 <b>and </b><br/><a NAME="E40:18_2"/><i><font color="Green" title="E51">A70</font></i>: 
<font color="Maroon" title="c30">Fk'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">a2'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c31">a2''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E36:18_2"><i><font color="Green" title="E48">A67</font></i></a></i>;<br/>
<a NAME="E41:18_2"/>
<font color="Maroon" title="c27">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E14:18_2"><i><font color="Green" title="E30">A49</font></i></a>, <a class="txt" href="polynom6.html#E24:18_2"><i><font color="Green" title="E38">A57</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E42:18_2"/><b>then </b><i><font color="Green" title="E52">A71</font></i>: 
<font color="Maroon" title="c16">Fn'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">j1</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E5:18_2"><i><font color="Green" title="E26">A45</font></i></a>, <a class="txt" href="polynom6.html#E15:18_2"><i><font color="Green" title="E31">A50</font></i></a>, <a class="txt" href="polynom6.html#E17:18_2"><i><font color="Green" title="E33">A52</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E43:18_2"/>
<font color="Maroon" title="c30">Fk'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E19:18_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="txt" href="polynom6.html#E33:18_2"><i><font color="Green" title="E45">A64</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E44:18_2"/><b>then </b>
<font color="Maroon" title="c17">Fm'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">Fk'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:18_2"><i><font color="Green" title="E27">A46</font></i></a>, <a class="txt" href="polynom6.html#E20:18_2"><i><font color="Green" title="E35">A54</font></i></a>, <a class="txt" href="polynom6.html#E22:18_2"><i><font color="Green" title="E37">A56</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E45:18_2"/><b>then </b><i><font color="Green" title="E53">A72</font></i>: 
( <font color="Maroon" title="c26">a1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">a1</font> &amp; <font color="Maroon" title="c28">a1''</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">a2</font> &amp; <font color="Maroon" title="c29">a2'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c20">c1</font> &amp; <font color="Maroon" title="c31">a2''</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c21">c2</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E10:18_2"><i><font color="Green" title="E28">A47</font></i></a>, <a class="txt" href="polynom6.html#E12:18_2"><i><font color="Green" title="E29">A48</font></i></a>, <a class="txt" href="polynom6.html#E31:18_2"><i><font color="Green" title="E44">A63</font></i></a>, <a class="txt" href="polynom6.html#E40:18_2"><i><font color="Green" title="E51">A70</font></i></a>, <a class="txt" href="polynom6.html#E42:18_2"><i><font color="Green" title="E52">A71</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<a NAME="E46:18_2"/><b>then </b><i><font color="Green" title="E54">A73</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a1</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E14:18_2"><i><font color="Green" title="E30">A49</font></i></a>, <a class="txt" href="polynom6.html#E25:18_2"><i><font color="Green" title="E39">A58</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E47:18_2"/><i><font color="Green" title="E55">A74</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">c1</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E19:18_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="txt" href="polynom6.html#E34:18_2"><i><font color="Green" title="E46">A65</font></i></a>, <a class="txt" href="polynom6.html#E45:18_2"><i><font color="Green" title="E53">A72</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E48:18_2"/>
 <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="c27">Fk</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="c27">Fk</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E49:18_2"/><b>then </b><i><font color="Green" title="E56">A75</font></i>: 
<font color="Maroon" title="c23">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E26:18_2"><i><font color="Green" title="E40">A59</font></i></a>, <a class="txt" href="polynom6.html#E28:18_2"><i><font color="Green" title="E42">A61</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="E50:18_2"/><b>then </b><i><font color="Green" title="E57">A76</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E30:18_2"><i><font color="Green" title="E43">A62</font></i></a>, <a class="txt" href="polynom6.html#E45:18_2"><i><font color="Green" title="E53">A72</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E51:18_2"/>
 <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="c30">Fk'</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="c30">Fk'</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E52:18_2"/><b>then </b><i><font color="Green" title="E58">A77</font></i>: 
<font color="Maroon" title="c25">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E35:18_2"><i><font color="Green" title="E47">A66</font></i></a>, <a class="txt" href="polynom6.html#E37:18_2"><i><font color="Green" title="E49">A68</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="E53:18_2"/><b>then </b><i><font color="Green" title="E59">A78</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E39:18_2"><i><font color="Green" title="E50">A69</font></i></a>, <a class="txt" href="polynom6.html#E45:18_2"><i><font color="Green" title="E53">A72</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c32">S</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font></span>)</span><b> such that </b><br/><a NAME="E2:18_2_2"/><i><font color="Green" title="E60">A79</font></i>: 
 <a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c1">o1</font></span>)</span>,<font color="Maroon" title="c32">S</font>
 <b>and </b><br/><a NAME="E3:18_2_2"/>
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> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">S</font> iff <font color="Olive" title="b1">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b1</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#D18" title="POLYNOM1:def.18">POLYNOM1:def 18</a></i>;<br/><a NAME="E4:18_2_2"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c1">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/><a NAME="E5:18_2_2"/><b>then </b><i><font color="Green" title="E61">A80</font></i>: 
 <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c1">o1</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c32">S</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T133" title="ORDERS_1:th.133">ORDERS_1:133</a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/><b>consider </b><font color="Maroon" title="c33">T</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font></span>)</span><b> such that </b><br/><a NAME="E7:18_2_2"/><i><font color="Green" title="E62">A81</font></i>: 
 <a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c33">T</font>
 <b>and </b><br/><a NAME="E8:18_2_2"/>
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="c2">o2</font> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">T</font> iff <font color="Olive" title="b1">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">b2</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#D18" title="POLYNOM1:def.18">POLYNOM1:def 18</a></i>;<br/><a NAME="E9:18_2_2"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c2">o2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/><a NAME="E10:18_2_2"/><b>then </b><i><font color="Green" title="E63">A82</font></i>: 
 <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c2">o2</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c33">T</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T133" title="ORDERS_1:th.133">ORDERS_1:133</a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/><div><i><font color="Green" title="E64">A83</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:18_2_2_1"/><i><font color="Green" title="E65">A84</font></i>: 
not <font color="Maroon" title="c22">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">i2</font>
 <b>and </b><br/><a NAME="E2:18_2_2_1"/><i><font color="Green" title="E66">A85</font></i>: 
(  not <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2</font> or  not <font color="Maroon" title="c23">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_2_1_1"/>
( <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2</font> or <font color="Maroon" title="c22">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">i2</font> )
 </span><b>by </b><i><a class="txt" href="polynom6.html#E1:18_2_2_1"><i><font color="Green" title="E65">A84</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:18_2_2_1_1_1"/>
<font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:18_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom6.html#E2:18_2"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="polynom6.html#E16:18_2"><i><font color="Green" title="E32">A51</font></i></a>, <a class="txt" href="polynom6.html#E21:18_2"><i><font color="Green" title="E36">A55</font></i></a>, <a class="txt" href="polynom6.html#E2:18_2_2_1"><i><font color="Green" title="E66">A85</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:18_2_2_1_1_2"/><i><font color="Green" title="E67">A86</font></i>: 
<font color="Maroon" title="c22">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:18_2_2_1_1_2"/><i><font color="Green" title="E68">A87</font></i>: 
<font color="Maroon" title="c25">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c24">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E20:18_2"><i><font color="Green" title="E35">A54</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:18_2_2_1_1_2"/><i><font color="Green" title="E69">A88</font></i>: 
 <a href="polynom1.html#K5" title="POLYNOM1:func.5">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> <font color="Maroon" title="c24">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c24">i2</font>
 <b>by </b><i><a class="ref" href="polynom3.html#T16" title="POLYNOM3:th.16">POLYNOM3:16</a></i>;<br/><a NAME="E4:18_2_2_1_1_2"/><i><font color="Green" title="E70">A89</font></i>: 
 <a href="polynom1.html#K5" title="POLYNOM1:func.5">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="c22">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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#K5" title="POLYNOM1:func.5">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="c22">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="E5:18_2_2_1_1_2"/>
<font color="Maroon" title="c24">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18_2_2_1_1_2"><i><font color="Green" title="E67">A86</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E6:18_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E71">A90</font></i>: 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:18_2_2_1_1_2"><i><font color="Green" title="E69">A88</font></i></a>, <a class="txt" href="polynom6.html#E4:18_2_2_1_1_2"><i><font color="Green" title="E70">A89</font></i></a>, <a class="ref" href="polynom3.html#T18" title="POLYNOM3:th.18">POLYNOM3:18</a></i>;<br/><a NAME="E7:18_2_2_1_1_2"/>
<font color="Maroon" title="c24">i2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="polynom6.html#E19:18_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:18_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E72">A91</font></i>: 
<font color="Maroon" title="c24">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E9:18_2_2_1_1_2"/>
<font color="Maroon" title="c24">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E10:18_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E73">A92</font></i>: 
<font color="Maroon" title="c24">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">i2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E11:18_2_2_1_1_2"/>
<font color="Maroon" title="c24">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E19:18_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:18_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E74">A93</font></i>: 
<font color="Maroon" title="c24">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:18_2_2_1_1_2"><i><font color="Green" title="E72">A91</font></i></a>, <a class="txt" href="polynom6.html#E10:18_2_2_1_1_2"><i><font color="Green" title="E73">A92</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:18_2_2_1_1_2"/>
<font color="Maroon" title="c24">i2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="polynom6.html#E19:18_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:18_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E75">A94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c34">G1</font> = <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i2</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#K13" title="POLYNOM1:func.13">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="txt" href="polynom6.html#E19:18_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="txt" href="polynom6.html#E33:18_2"><i><font color="Green" title="E45">A64</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c35">GG1</font> = <span class="p1"><a href="matrlin.html#K3" title="MATRLIN:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c34">G1</font></span><a href="matrlin.html#K3" title="MATRLIN:func.3">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">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>  ;<br/><a NAME="E17:18_2_2_1_1_2"/>
<font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c24">i2</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="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c35">GG1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E12:18_2_2_1_1_2"><i><font color="Green" title="E74">A93</font></i></a>, <a class="txt" href="polynom6.html#E14:18_2_2_1_1_2"><i><font color="Green" title="E75">A94</font></i></a>, <a class="ref" href="polynom3.html#T19" title="POLYNOM3:th.19">POLYNOM3:19</a></i>;<br/><a NAME="E18:18_2_2_1_1_2"/><b>then </b>
 <a href="polynom1.html#K5" title="POLYNOM1:func.5">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> <font color="Maroon" title="c24">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c35">GG1</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T13" title="POLYNOM1:th.13">POLYNOM1:13</a></i>;<br/><a NAME="E19:18_2_2_1_1_2"/><b>then </b>
 <a href="polynom1.html#K5" title="POLYNOM1:func.5">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> <font color="Maroon" title="c24">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c34">G1</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#T12" title="POLYNOM1:th.12">POLYNOM1:12</a></i>;<br/><a NAME="E20:18_2_2_1_1_2"/><b>then </b>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <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#K5" title="POLYNOM1:func.5">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="c24">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</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="c24">i2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>;<br/><a NAME="E21:18_2_2_1_1_2"/><b>then </b>
<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#K5" title="POLYNOM1:func.5">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="c24">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:18_2_2_1_1_2"><i><font color="Green" title="E68">A87</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E22:18_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E76">A95</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">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="c24">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E6:18_2_2_1_1_2"><i><font color="Green" title="E71">A90</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E23:18_2_2_1_1_2"/>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#K5" title="POLYNOM1:func.5">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="c22">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c23">j1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E24:18_2_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom6.html#E2:18_2"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="polynom6.html#E16:18_2"><i><font color="Green" title="E32">A51</font></i></a>, <a class="txt" href="polynom6.html#E21:18_2"><i><font color="Green" title="E36">A55</font></i></a>, <a class="txt" href="polynom6.html#E22:18_2_2_1_1_2"><i><font color="Green" title="E76">A95</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_2_2"/>
( <font color="Maroon" title="c22">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">i2</font> or ( <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2</font> &amp; <font color="Maroon" title="c23">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="polynom6.html#E11:18_2_2"><i><font color="Green" title="E64">A83</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_2_1"><b>case </b></a><a NAME="E1:18_2_2_2_1"/><i><font color="Green" title="E65">A96</font></i>: 
<font color="Maroon" title="c22">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">a1</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c20">c1</font></span><br/><div class="add"><a NAME="E2:18_2_2_2_1"/><b>then </b><i><font color="Green" title="E66">A97</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c22">i1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c24">i2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c1">o1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E14:18_2"><i><font color="Green" title="E30">A49</font></i></a>, <a class="txt" href="polynom6.html#E19:18_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="txt" href="polynom6.html#E2:18_2_2"><i><font color="Green" title="E60">A79</font></i></a>, <a class="txt" href="polynom6.html#E5:18_2_2"><i><font color="Green" title="E61">A80</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><a NAME="E3:18_2_2_2_1"/><i><font color="Green" title="E67">A98</font></i>: 
<font color="Maroon" title="c18">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">c1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E14:18_2"><i><font color="Green" title="E30">A49</font></i></a>, <a class="txt" href="polynom6.html#E19:18_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="txt" href="polynom6.html#E46:18_2"><i><font color="Green" title="E54">A73</font></i></a>, <a class="txt" href="polynom6.html#E47:18_2"><i><font color="Green" title="E55">A74</font></i></a>, <a class="txt" href="polynom6.html#E1:18_2_2_2_1"><i><font color="Green" title="E65">A96</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E4:18_2_2_2_1"/>
<font color="Maroon" title="c18">a1</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c20">c1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E25:18_2"><i><font color="Green" title="E39">A58</font></i></a>, <a class="txt" href="polynom6.html#E34:18_2"><i><font color="Green" title="E46">A65</font></i></a>, <a class="txt" href="polynom6.html#E45:18_2"><i><font color="Green" title="E53">A72</font></i></a>, <a class="txt" href="polynom6.html#E2:18_2_2_2_1"><i><font color="Green" title="E66">A97</font></i></a>, <a class="ref" href="polynom1.html#D16" title="POLYNOM1:def.16">POLYNOM1:def 16</a></i>;<br/><b>hence </b><a NAME="E5:18_2_2_2_1"/>
<font color="Maroon" title="c18">a1</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c20">c1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:18_2_2_2_1"><i><font color="Green" title="E67">A98</font></i></a>, <a class="ref" href="polynom1.html#D12" title="POLYNOM1:def.12">POLYNOM1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_2_2"><b>case </b></a><b>that </b><a NAME="E1:18_2_2_2_2"/><i><font color="Green" title="E65">A99</font></i>: 
<font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i2</font>
 <b>and </b><br/><a NAME="E2:18_2_2_2_2"/><i><font color="Green" title="E66">A100</font></i>: 
<font color="Maroon" title="c23">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">a1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c20">c1</font> &amp; <font color="Maroon" title="c19">a2</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c21">c2</font> )</span><br/><div class="add"><a NAME="E3:18_2_2_2_2"/><i><font color="Green" title="E67">A101</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c23">j1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c25">j2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c2">o2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E49:18_2"><i><font color="Green" title="E56">A75</font></i></a>, <a class="txt" href="polynom6.html#E52:18_2"><i><font color="Green" title="E58">A77</font></i></a>, <a class="txt" href="polynom6.html#E7:18_2_2"><i><font color="Green" title="E62">A81</font></i></a>, <a class="txt" href="polynom6.html#E10:18_2_2"><i><font color="Green" title="E63">A82</font></i></a>, <a class="txt" href="polynom6.html#E2:18_2_2_2_2"><i><font color="Green" title="E66">A100</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><b>thus </b><a NAME="E4:18_2_2_2_2"/>
<font color="Maroon" title="c18">a1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c20">c1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E25:18_2"><i><font color="Green" title="E39">A58</font></i></a>, <a class="txt" href="polynom6.html#E34:18_2"><i><font color="Green" title="E46">A65</font></i></a>, <a class="txt" href="polynom6.html#E45:18_2"><i><font color="Green" title="E53">A72</font></i></a>, <a class="txt" href="polynom6.html#E1:18_2_2_2_2"><i><font color="Green" title="E65">A99</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="c19">a2</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c21">c2</font></span><br/><a NAME="E5:18_2_2_2_2"/><i><font color="Green" title="E68">A102</font></i>: 
<font color="Maroon" title="c19">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">c2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E49:18_2"><i><font color="Green" title="E56">A75</font></i></a>, <a class="txt" href="polynom6.html#E50:18_2"><i><font color="Green" title="E57">A76</font></i></a>, <a class="txt" href="polynom6.html#E52:18_2"><i><font color="Green" title="E58">A77</font></i></a>, <a class="txt" href="polynom6.html#E53:18_2"><i><font color="Green" title="E59">A78</font></i></a>, <a class="txt" href="polynom6.html#E2:18_2_2_2_2"><i><font color="Green" title="E66">A100</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E6:18_2_2_2_2"/>
<font color="Maroon" title="c19">a2</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c21">c2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E30:18_2"><i><font color="Green" title="E43">A62</font></i></a>, <a class="txt" href="polynom6.html#E39:18_2"><i><font color="Green" title="E50">A69</font></i></a>, <a class="txt" href="polynom6.html#E45:18_2"><i><font color="Green" title="E53">A72</font></i></a>, <a class="txt" href="polynom6.html#E3:18_2_2_2_2"><i><font color="Green" title="E67">A101</font></i></a>, <a class="ref" href="polynom1.html#D16" title="POLYNOM1:def.16">POLYNOM1:def 16</a></i>;<br/><b>hence </b><a NAME="E7:18_2_2_2_2"/>
<font color="Maroon" title="c19">a2</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c21">c2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:18_2_2_2_2"><i><font color="Green" title="E68">A102</font></i></a>, <a class="ref" href="polynom1.html#D12" title="POLYNOM1:def.12">POLYNOM1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E55:18_2"/><b>then </b>
( <font color="Maroon" title="c14">Fn</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c15">Fm</font> or <font color="Maroon" title="c14">Fn</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">Fm</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E10:18_2"><i><font color="Green" title="E28">A47</font></i></a>, <a class="txt" href="polynom6.html#E12:18_2"><i><font color="Green" title="E29">A48</font></i></a>, <a class="ref" href="polynom6.html#T11" target="_self" title="POLYNOM6:th.11">Th11</a></i>;<br/>
<a NAME="E56:18_2"/><b>then </b>
<font color="Maroon" title="c14">Fn</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c15">Fm</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D12" title="POLYNOM1:def.12">POLYNOM1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E57:18_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</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="ref" href="polynom1.html#D16" title="POLYNOM1:def.16">POLYNOM1:def 16</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="E23:18"/><b>then </b><i><font color="Green" title="E24">A103</font></i>: 
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</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>,<font color="Maroon" title="c10">S</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E21:18"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E24:18"/>
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> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font> iff <font color="Olive" title="b1">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c12">T</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font></span>)</span><b> such that </b><br/><a NAME="E2:18_3"/><i><font color="Green" title="E25">A104</font></i>: 
 <a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c1">o1</font></span>)</span>,<font color="Maroon" title="c12">T</font>
 <b>and </b><br/><a NAME="E3:18_3"/><i><font color="Green" title="E26">A105</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> holds <br/> ( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">T</font> iff <font color="Olive" title="b1">q</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b1</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#D18" title="POLYNOM1:def.18">POLYNOM1:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">W</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font></span>)</span><b> such that </b><br/><a NAME="E5:18_3"/><i><font color="Green" title="E27">A106</font></i>: 
 <a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c13">W</font>
 <b>and </b><br/><a NAME="E6:18_3"/><i><font color="Green" title="E28">A107</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font> holds <br/> ( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">W</font> iff <font color="Olive" title="b1">q</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">b2</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#D18" title="POLYNOM1:def.18">POLYNOM1:def 18</a></i>;<br/>
<a NAME="E7:18_3"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c1">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/>
<a NAME="E8:18_3"/><b>then </b>
 <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c1">o1</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c12">T</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T133" title="ORDERS_1:th.133">ORDERS_1:133</a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/>
<a NAME="E9:18_3"/><b>then </b><i><font color="Green" title="E29">A108</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c1">o1</font></span>)</span>,<font color="Maroon" title="c12">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">T</font>
 
<b>by </b><i><a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E10:18_3"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c2">o2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/>
<a NAME="E11:18_3"/><b>then </b>
 <a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c2">o2</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c13">W</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T133" title="ORDERS_1:th.133">ORDERS_1:133</a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/>
<a NAME="E12:18_3"/><b>then </b><i><font color="Green" title="E30">A109</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">BagOrder</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c13">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">W</font>
 
<b>by </b><i><a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<b>let </b><font color="Maroon" title="c14">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="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font> iff <font color="Maroon" title="c14">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <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><br/>

<b>thus </b><a NAME="E13:18_3"/>
( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font> implies <font color="Maroon" title="c14">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font> implies <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3_1"/>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <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><br/>

<b>then consider </b><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:18_3_1"/><i><font color="Green" title="E31">A110</font></i>: 
<font color="Maroon" title="c15">x</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="c5">G</font></span>)</span>
 <b>and </b><br/><a NAME="E4:18_3_1"/><i><font color="Green" title="E32">A111</font></i>: 
<font color="Maroon" title="c14">p</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="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">i</font>, <font color="Maroon" title="c17">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:18_3_1"/><i><font color="Green" title="E33">A112</font></i>: 
<font color="Maroon" title="c16">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="E7:18_3_1"/><i><font color="Green" title="E34">A113</font></i>: 
<font color="Maroon" title="c17">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="c16">i</font></span>)</span>
 <b>and </b><br/><a NAME="E8:18_3_1"/>
<font color="Maroon" title="c15">x</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#K5" title="POLYNOM1:func.5">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="c16">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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="c17">j</font>
 <b>and </b><br/><a NAME="E9:18_3_1"/><i><font color="Green" title="E35">A114</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="c16">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:18_3_1"><i><font color="Green" title="E31">A110</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">a1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c19">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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="E11:18_3_1"/><i><font color="Green" title="E36">A115</font></i>: 
<font color="Maroon" title="c19">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="Maroon" title="c16">i</font>
 <b>and </b><br/><a NAME="E12:18_3_1"/><i><font color="Green" title="E37">A116</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a1'</font>
 <b>and </b><br/><a NAME="E13:18_3_1"/><i><font color="Green" title="E38">A117</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">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">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E14:18_3_1"/><i><font color="Green" title="E39">A118</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="c19">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#K13" title="POLYNOM1:func.13">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">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="c19">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="c18">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="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="polynom6.html#E6:18_3_1"><i><font color="Green" title="E33">A112</font></i></a></i>;<br/>
<a NAME="E15:18_3_1"/><i><font color="Green" title="E40">A119</font></i>: 
<font color="Maroon" title="c19">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:18_3_1"><i><font color="Green" title="E33">A112</font></i></a>, <a class="txt" href="polynom6.html#E11:18_3_1"><i><font color="Green" title="E36">A115</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="c20">a1''</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E17:18_3_1"/><i><font color="Green" title="E41">A120</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1''</font>
 <b>and </b><br/><a NAME="E18:18_3_1"/><i><font color="Green" title="E42">A121</font></i>: 
<font color="Maroon" title="c19">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c20">a1''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E7:18_3_1"><i><font color="Green" title="E34">A113</font></i></a>, <a class="txt" href="polynom6.html#E14:18_3_1"><i><font color="Green" title="E39">A118</font></i></a></i>;<br/>
<a NAME="E19:18_3_1"/><i><font color="Green" title="E43">A122</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c20">a1''</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:18_3_1"><i><font color="Green" title="E32">A111</font></i></a>, <a class="txt" href="polynom6.html#E7:18_3_1"><i><font color="Green" title="E34">A113</font></i></a>, <a class="txt" href="polynom6.html#E9:18_3_1"><i><font color="Green" title="E35">A114</font></i></a>, <a class="txt" href="polynom6.html#E15:18_3_1"><i><font color="Green" title="E40">A119</font></i></a>, <a class="txt" href="polynom6.html#E18:18_3_1"><i><font color="Green" title="E42">A121</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E20:18_3_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a1'</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E6:18_3_1"><i><font color="Green" title="E33">A112</font></i></a>, <a class="txt" href="polynom6.html#E12:18_3_1"><i><font color="Green" title="E37">A116</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E21:18_3_1"/><b>then </b><i><font color="Green" title="E44">A123</font></i>: 
<font color="Maroon" title="c18">a1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">T</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E2:18_3"><i><font color="Green" title="E25">A104</font></i></a>, <a class="txt" href="polynom6.html#E9:18_3"><i><font color="Green" title="E29">A108</font></i></a>, <a class="txt" href="polynom6.html#E6:18_3_1"><i><font color="Green" title="E33">A112</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">a11'</font> = <font color="Maroon" title="c18">a1'</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> ;<br/>
<a NAME="E23:18_3_1"/><i><font color="Green" title="E45">A124</font></i>: 
<font color="Maroon" title="c21">a11'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:18_3"><i><font color="Green" title="E26">A105</font></i></a>, <a class="txt" href="polynom6.html#E21:18_3_1"><i><font color="Green" title="E44">A123</font></i></a></i>;<br/>
<a NAME="E24:18_3_1"/>
<font color="Maroon" title="c17">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> <font color="Maroon" title="c19">Fk</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E7:18_3_1"><i><font color="Green" title="E34">A113</font></i></a>, <a class="txt" href="polynom6.html#E15:18_3_1"><i><font color="Green" title="E40">A119</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="E25:18_3_1"/><b>then </b><i><font color="Green" title="E46">A125</font></i>: 
<font color="Maroon" title="c17">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">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E13:18_3_1"><i><font color="Green" title="E38">A117</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="E26:18_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1''</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E17:18_3_1"><i><font color="Green" title="E41">A120</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E27:18_3_1"/><b>then </b><i><font color="Green" title="E47">A126</font></i>: 
<font color="Maroon" title="c20">a1''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">W</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E5:18_3"><i><font color="Green" title="E27">A106</font></i></a>, <a class="txt" href="polynom6.html#E12:18_3"><i><font color="Green" title="E30">A109</font></i></a>, <a class="txt" href="polynom6.html#E25:18_3_1"><i><font color="Green" title="E46">A125</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">a11''</font> = <font color="Maroon" title="c20">a1''</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font> ;<br/>
<a NAME="E29:18_3_1"/>
<font color="Maroon" title="c22">a11''</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">b2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:18_3"><i><font color="Green" title="E28">A107</font></i></a>, <a class="txt" href="polynom6.html#E27:18_3_1"><i><font color="Green" title="E47">A126</font></i></a></i>;<br/>
<b>hence </b><a NAME="E30:18_3_1"/>
<font color="Maroon" title="c14">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E19:18_3_1"><i><font color="Green" title="E43">A122</font></i></a>, <a class="txt" href="polynom6.html#E23:18_3_1"><i><font color="Green" title="E45">A124</font></i></a>, <a class="ref" href="polynom6.html#T9" target="_self" title="POLYNOM6:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E14:18_3"/>
( <font color="Maroon" title="c14">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font> implies <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3_2"/>
<font color="Maroon" title="c14">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c4">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">p1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c16">p2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E3:18_3_2"/><i><font color="Green" title="E31">A127</font></i>: 
<font color="Maroon" title="c15">p1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b1</font>
 <b>and </b><br/><a NAME="E4:18_3_2"/><i><font color="Green" title="E32">A128</font></i>: 
<font color="Maroon" title="c16">p2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">b2</font>
 <b>and </b><br/><a NAME="E5:18_3_2"/><i><font color="Green" title="E33">A129</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">p1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c16">p2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#T10" target="_self" title="POLYNOM6:th.10">Th10</a></i>;<br/>
<a NAME="E6:18_3_2"/>
<font color="Maroon" title="c15">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">T</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:18_3"><i><font color="Green" title="E26">A105</font></i></a>, <a class="txt" href="polynom6.html#E3:18_3_2"><i><font color="Green" title="E31">A127</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:18_3_2"/><i><font color="Green" title="E34">A130</font></i>: 
<font color="Maroon" title="c17">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">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E9:18_3_2"/><i><font color="Green" title="E35">A131</font></i>: 
<font color="Maroon" title="c15">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:18_3"><i><font color="Green" title="E25">A104</font></i></a>, <a class="txt" href="polynom6.html#E9:18_3"><i><font color="Green" title="E29">A108</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:18_3_2"/>
<font color="Maroon" title="c16">p2</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> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E5:18_3"><i><font color="Green" title="E27">A106</font></i></a>, <a class="txt" href="polynom6.html#E6:18_3"><i><font color="Green" title="E28">A107</font></i></a>, <a class="txt" href="polynom6.html#E12:18_3"><i><font color="Green" title="E30">A109</font></i></a>, <a class="txt" href="polynom6.html#E4:18_3_2"><i><font color="Green" title="E32">A128</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:18_3_2"/><i><font color="Green" title="E36">A132</font></i>: 
<font color="Maroon" title="c18">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">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E13:18_3_2"/><i><font color="Green" title="E37">A133</font></i>: 
<font color="Maroon" title="c16">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">i</font> = <font color="Maroon" title="c17">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="polynom6.html#E8:18_3_2"><i><font color="Green" title="E34">A130</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">a1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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#K13" title="POLYNOM1:func.13">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="E16:18_3_2"/><i><font color="Green" title="E38">A134</font></i>: 
<font color="Maroon" title="c21">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="Maroon" title="c19">i</font>
 <b>and </b><br/><a NAME="E17:18_3_2"/><i><font color="Green" title="E39">A135</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1'</font>
 <b>and </b><br/><a NAME="E18:18_3_2"/><i><font color="Green" title="E40">A136</font></i>: 
 <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#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E19:18_3_2"/><i><font color="Green" title="E41">A137</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">a1''</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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">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="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">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="polynom6.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="polynom6.html#E8:18_3_2"><i><font color="Green" title="E34">A130</font></i></a></i>;<br/>
<a NAME="E20:18_3_2"/><i><font color="Green" title="E42">A138</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E12:18_3_2"><i><font color="Green" title="E36">A132</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="E21:18_3_2"/>
 <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="c19">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E8:18_3_2"><i><font color="Green" title="E34">A130</font></i></a>, <a class="txt" href="polynom6.html#E16:18_3_2"><i><font color="Green" title="E38">A134</font></i></a>, <a class="txt" href="polynom6.html#E18:18_3_2"><i><font color="Green" title="E40">A136</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E22:18_3_2"/><b>then </b><i><font color="Green" title="E43">A139</font></i>: 
<font color="Maroon" title="c18">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="c19">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E20:18_3_2"><i><font color="Green" title="E42">A138</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">j</font> = <font color="Maroon" title="c18">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="polynom6.html#E12:18_3_2"><i><font color="Green" title="E36">A132</font></i></a></i>;<br/>
<a NAME="E24:18_3_2"/><i><font color="Green" title="E44">A140</font></i>: 
( <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">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="c19">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> &amp; <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="c19">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p3">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E8:18_3_2"><i><font color="Green" title="E34">A130</font></i></a>, <a class="txt" href="polynom6.html#E22:18_3_2"><i><font color="Green" title="E43">A139</font></i></a>, <a class="ref" href="polynom1.html#T33" title="POLYNOM1:th.33">POLYNOM1:33</a></i>;<br/>
<a NAME="E25:18_3_2"/><i><font color="Green" title="E45">A141</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="polynom6.html#E8:18_3_2"><i><font color="Green" title="E34">A130</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="c23">a1''</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E27:18_3_2"/><i><font color="Green" title="E46">A142</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a1''</font>
 <b>and </b><br/><a NAME="E28:18_3_2"/><i><font color="Green" title="E47">A143</font></i>: 
<font color="Maroon" title="c21">Fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c23">a1''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E16:18_3_2"><i><font color="Green" title="E38">A134</font></i></a>, <a class="txt" href="polynom6.html#E19:18_3_2"><i><font color="Green" title="E41">A137</font></i></a>, <a class="txt" href="polynom6.html#E22:18_3_2"><i><font color="Green" title="E43">A139</font></i></a></i>;<br/>
<a NAME="E29:18_3_2"/><i><font color="Green" title="E48">A144</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="c19">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c23">a1''</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E16:18_3_2"><i><font color="Green" title="E38">A134</font></i></a>, <a class="txt" href="polynom6.html#E22:18_3_2"><i><font color="Green" title="E43">A139</font></i></a>, <a class="txt" href="polynom6.html#E25:18_3_2"><i><font color="Green" title="E45">A141</font></i></a>, <a class="txt" href="polynom6.html#E28:18_3_2"><i><font color="Green" title="E47">A143</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E30:18_3_2"/><i><font color="Green" title="E49">A145</font></i>: 
<font color="Maroon" title="c20">a1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">p1</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E8:18_3_2"><i><font color="Green" title="E34">A130</font></i></a>, <a class="txt" href="polynom6.html#E9:18_3_2"><i><font color="Green" title="E35">A131</font></i></a>, <a class="txt" href="polynom6.html#E17:18_3_2"><i><font color="Green" title="E39">A135</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E31:18_3_2"/>
<font color="Maroon" title="c23">a1''</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">p2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E12:18_3_2"><i><font color="Green" title="E36">A132</font></i></a>, <a class="txt" href="polynom6.html#E13:18_3_2"><i><font color="Green" title="E37">A133</font></i></a>, <a class="txt" href="polynom6.html#E27:18_3_2"><i><font color="Green" title="E46">A142</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="E32:18_3_2"/>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:18_3_2"><i><font color="Green" title="E33">A129</font></i></a>, <a class="txt" href="polynom6.html#E24:18_3_2"><i><font color="Green" title="E44">A140</font></i></a>, <a class="txt" href="polynom6.html#E29:18_3_2"><i><font color="Green" title="E48">A144</font></i></a>, <a class="txt" href="polynom6.html#E30:18_3_2"><i><font color="Green" title="E49">A145</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:18"/>
 <a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c5">G</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E23:18"><i><font color="Green" title="E24">A103</font></i></a>, <a class="ref" href="polynom1.html#D18" title="POLYNOM1:def.18">POLYNOM1:def 18</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>
