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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<br/><b>let </b><font color="Maroon" title="c4">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>;<br/>









<b>set </b><font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">q'</font> = <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c3">p</font>;<br/>
<b>set </b><font color="Maroon" title="c8">m</font> =  <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c5">a</font> as  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">non-zero</a> <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></i>;<br/><i><font color="Green" title="E30">A1</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> = 
 <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c3">p</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_1_2"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c10">u</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">t</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E4:35_1_1_2"/><i><font color="Green" title="E31">A2</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c12">t</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">t</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#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">t</font> holds <br/> ex <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c12">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b3">b2</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></i>;<br/><a NAME="E5:35_1_1_2"/><i><font color="Green" title="E32">A3</font></i>: 
 <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="polynom7.html#T10" title="POLYNOM7:th.10">POLYNOM7:10</a></i>;<br/><i><font color="Green" title="E33">A4</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">t</font> = 
 <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#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E4:35_1_1_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_2_2_1"/>
( <font color="Maroon" title="c4">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c11">s</font> or  not <font color="Maroon" title="c4">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c11">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:35_1_1_2_2_1_1"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c11">s</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">s'</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:35_1_1_2_2_1_1"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c13">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font>
 <b>by </b><i><a class="ref" href="termord.html#T1" title="TERMORD:th.1">TERMORD:1</a></i>;<br/><i><font color="Green" title="E36">A7</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">s</font> = 
<font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">s</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D10" title="POLYNOM7:def.10">POLYNOM7:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E1:35_1_1_2_2_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="polyred.html#D1" target="_self" title="POLYRED:def.1">Def1</a></i>
;<br/><b>consider </b><font color="Maroon" title="c14">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:35_1_1_2_2_1_1"/><i><font color="Green" title="E37">A8</font></i>: 
( <font color="Maroon" title="c14">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#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c13">s'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polyred.html#E3:35_1_1_2_2_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="polynom1.html#T73" title="POLYNOM1:th.73">POLYNOM1:73</a></i>;<br/><a NAME="E7:35_1_1_2_2_1_1"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s'</font>
 <b>by </b><i><a class="txt" href="polyred.html#E3:35_1_1_2_2_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c16">b2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E9:35_1_1_2_2_1_1"/><i><font color="Green" title="E39">A10</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c12">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c15">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c16">b2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polyred.html#E4:35_1_1_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="polyred.html#E6:35_1_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="polyred.html#E6:35_1_1_2_2_1_1"><i><font color="Green" title="E37">A8</font></i></a></i>;<br/><i><font color="Green" title="E40">A11</font></i>: <font color="Maroon" title="c15">b1</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c13">s'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="polyred.html#E6:35_1_1_2_2_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="polyred.html#E9:35_1_1_2_2_1_1"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c4">b</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><i><font color="Green" title="E41">A12</font></i>: <font color="Maroon" title="c16">b2</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c13">s'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="polyred.html#E6:35_1_1_2_2_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="polyred.html#E9:35_1_1_2_2_1_1"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c13">s'</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">i'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:35_1_1_2_2_1_1_4"/><i><font color="Green" title="E42">A13</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> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c17">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">i</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c18">b1'</font>, <font color="Maroon" title="c19">b2'</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:35_1_1_2_2_1_1_4"/><i><font color="Green" title="E43">A14</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c18">b1'</font>,<font color="Maroon" title="c19">b2'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c12">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c18">b1'</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c19">b2'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polyred.html#E4:35_1_1_2"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c20">h1</font>, <font color="Maroon" title="c21">h2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:35_1_1_2_2_1_1_4"/><i><font color="Green" title="E44">A15</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c20">h1</font>,<font color="Maroon" title="c21">h2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c21">h2</font> )
 <b>by </b><i><a class="txt" href="polyred.html#E6:35_1_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="polyred.html#E1:35_1_1_2_2_1_1_4"><i><font color="Green" title="E42">A13</font></i></a>, <a class="ref" href="polynom1.html#T72" title="POLYNOM1:th.72">POLYNOM1:72</a></i>;<br/><i><font color="Green" title="E45">A16</font></i>: <font color="Maroon" title="c20">h1</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c18">b1'</font>,<font color="Maroon" title="c19">b2'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="polyred.html#E3:35_1_1_2_2_1_1_4"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="polyred.html#E5:35_1_1_2_2_1_1_4"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c18">b1'</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><a NAME="E7:35_1_1_2_2_1_1_4"/><i><font color="Green" title="E46">A17</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c20">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font>
 <b>by </b><i><a class="txt" href="polyred.html#E5:35_1_1_2_2_1_1_4"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2_1_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1_2_2_1_1_4_2"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c18">b1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ;<br/><a NAME="E2:35_1_1_2_2_1_1_4_2"/><b>then </b>
<font color="Maroon" title="c18">b1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="polyred.html#E5:35_1_1_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="polynom7.html#D6" title="POLYNOM7:def.6">POLYNOM7:def 6</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i'</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="polyred.html#E6:35_1_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="polyred.html#E6:35_1_1_2_2_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="polyred.html#E7:35_1_1_2_2_1_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="polyred.html#E1:35_1_1_2_2_1_1_4"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="polyred.html#E5:35_1_1_2_2_1_1_4"><i><font color="Green" title="E44">A15</font></i></a>, <a class="txt" href="polyred.html#E6:35_1_1_2_2_1_1_4"><i><font color="Green" title="E45">A16</font></i></a>, <a class="txt" href="polyred.html#E7:35_1_1_2_2_1_1_4"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="polyred.html#E6:35_1_1_2_2_1_1"><i><font color="Green" title="E37">A8</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="E4:35_1_1_2_2_1_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="polyred.html#E6:35_1_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="polyred.html#E6:35_1_1_2_2_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="polyred.html#E1:35_1_1_2_2_1_1_4"><i><font color="Green" title="E42">A13</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:35_1_1_2_2_1_1_4"/>
<font color="Maroon" title="c12">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="polyred.html#E3:35_1_1_2_2_1_1_4"><i><font color="Green" title="E43">A14</font></i></a>, <a class="ref" href="binom.html#T1" title="BINOM:th.1">BINOM:1</a></i>;<br/></div><b>end;</b></div><b>then </b> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c12">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E6:35_1_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="polyred.html#E6:35_1_1_2_2_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="polyred.html#E7:35_1_1_2_2_1_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="polyred.html#E9:35_1_1_2_2_1_1"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="polyred.html#E10:35_1_1_2_2_1_1"><i><font color="Green" title="E40">A11</font></i></a>, <a class="txt" href="polyred.html#E11:35_1_1_2_2_1_1"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="polynom2.html#T5" title="POLYNOM2:th.5">POLYNOM2:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E5:35_1_1_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="polynom7.html#D7" title="POLYNOM7:def.7">POLYNOM7:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T9" title="POLYNOM7:th.9">POLYNOM7:9</a></i>
;<br/><b>hence </b><a NAME="E14:35_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">s</font>
 <b>by </b><i><a class="txt" href="polyred.html#E4:35_1_1_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="polyred.html#E4:35_1_1_2_2_1_1"><i><font color="Green" title="E36">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:35_1_1_2_2_1_2"/><i><font color="Green" title="E34">A18</font></i>: 
not <font color="Maroon" title="c4">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c11">s</font>
 ;<br/><div class="add"><i><font color="Green" title="E35">A19</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">s</font> = 
<font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">s</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D10" title="POLYNOM7:def.10">POLYNOM7:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E1:35_1_1_2_2_1_2"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="polyred.html#D1" target="_self" title="POLYRED:def.1">Def1</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="ref" href="binom.html#T2" title="BINOM:th.2">BINOM:2</a></i>
;<br/><b>consider </b><font color="Maroon" title="c13">t</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E4:35_1_1_2_2_1_2"/><i><font color="Green" title="E36">A20</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c13">t</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">t</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#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">t</font> holds <br/> ex <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c13">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b3">b2</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:35_1_1_2_2_1_2_2"/><i><font color="Green" title="E37">A21</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">t</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">b1'</font>, <font color="Maroon" title="c16">b2'</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:35_1_1_2_2_1_2_2"/><i><font color="Green" title="E38">A22</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c15">b1'</font>,<font color="Maroon" title="c16">b2'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c13">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c15">b1'</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c16">b2'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polyred.html#E4:35_1_1_2_2_1_2"><i><font color="Green" title="E36">A20</font></i></a></i>;<br/><i><font color="Green" title="E39">A23</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">t</font> = 
 <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#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E4:35_1_1_2_2_1_2"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2_1_2_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_2_2_1_2_2_2_1"/>
( <font color="Maroon" title="c15">b1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> or <font color="Maroon" title="c15">b1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_2_2_2_1_1"><b>case </b></a><a NAME="E1:35_1_1_2_2_1_2_2_2_1_1"/><i><font color="Green" title="E40">A24</font></i>: 
<font color="Maroon" title="c15">b1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c17">h1</font>, <font color="Maroon" title="c18">h2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:35_1_1_2_2_1_2_2_2_1_1"/><i><font color="Green" title="E41">A25</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">h1</font>,<font color="Maroon" title="c18">h2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">h1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c18">h2</font> )
 <b>by </b><i><a class="txt" href="polyred.html#E1:35_1_1_2_2_1_2_2"><i><font color="Green" title="E37">A21</font></i></a>, <a class="txt" href="polyred.html#E4:35_1_1_2_2_1_2_2"><i><font color="Green" title="E39">A23</font></i></a>, <a class="ref" href="polynom1.html#T72" title="POLYNOM1:th.72">POLYNOM1:72</a></i>;<br/><font color="Maroon" title="c17">h1</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c15">b1'</font>,<font color="Maroon" title="c16">b2'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="polyred.html#E3:35_1_1_2_2_1_2_2"><i><font color="Green" title="E38">A22</font></i></a>, <a class="txt" href="polyred.html#E3:35_1_1_2_2_1_2_2_2_1_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c15">b1'</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><b>hence </b><a NAME="E5:35_1_1_2_2_1_2_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polyred.html#E5:35_1_1_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="polyred.html#E1:35_1_1_2_2_1_2"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="polyred.html#E1:35_1_1_2_2_1_2_2_2_1_1"><i><font color="Green" title="E40">A24</font></i></a>, <a class="txt" href="polyred.html#E3:35_1_1_2_2_1_2_2_2_1_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="termord.html#T1" title="TERMORD:th.1">TERMORD:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_2_2_2_1_2"><b>case </b></a><a NAME="E1:35_1_1_2_2_1_2_2_2_1_2"/>
<font color="Maroon" title="c15">b1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_1_2_2_1_2_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c15">b1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="polyred.html#E14"><i><font color="Green" title="E14">Lm8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:35_1_1_2_2_1_2_2"/>
<font color="Maroon" title="c13">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="polyred.html#E3:35_1_1_2_2_1_2_2"><i><font color="Green" title="E38">A22</font></i></a>, <a class="ref" href="binom.html#T1" title="BINOM:th.1">BINOM:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:35_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font>
 <b>by </b><i><a class="txt" href="polyred.html#E2:35_1_1_2_2_1_2"><i><font color="Green" title="E35">A19</font></i></a>, <a class="txt" href="polyred.html#E4:35_1_1_2_2_1_2"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="matrlin.html#T15" title="MATRLIN:th.15">MATRLIN:15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:35_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:35_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="polyred.html#E3:35_1_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/><i><font color="Green" title="E30">A26</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ;<br/><div class="add"><i><font color="Green" title="E31">A27</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> = 
 <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_2_2"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">u'</font> = <font color="Maroon" title="c9">u</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">u'</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">u'</font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E1:35_1_2"><i><font color="Green" title="E30">A26</font></i></a>, <a class="ref" href="polynom7.html#D10" title="POLYNOM7:def.10">POLYNOM7:def 10</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="ref" href="binom.html#T1" title="BINOM:th.1">BINOM:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">u'</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>
;<br/><b>hence </b><a NAME="E4:35_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 ;<br/></div><b>end;</b></div><a NAME="E4:35_1_2"/><b>then </b><i><font color="Green" title="E32">A28</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="polyred.html#E2:35_1_2"><i><font color="Green" title="E31">A27</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><i><font color="Green" title="E33">A29</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> = 
 <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_2_4"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">u'</font> = <font color="Maroon" title="c9">u</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> <font color="Maroon" title="c1">n</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_2_4_1_1"/>
( <font color="Maroon" title="c10">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> or <font color="Maroon" title="c10">u'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_4_1_1_1"><b>case </b></a><a NAME="E1:35_1_2_4_1_1_1"/><i><font color="Green" title="E34">A30</font></i>: 
<font color="Maroon" title="c10">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:35_1_2_4_1_1_1"/><b>then </b>
( <font color="Maroon" title="c10">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> )
 <b>by </b><i><a class="txt" href="polyred.html#E1:35_1_2"><i><font color="Green" title="E30">A26</font></i></a>, <a class="ref" href="polynom7.html#T8" title="POLYNOM7:th.8">POLYNOM7:8</a></i>;<br/><a NAME="E3:35_1_2_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="polyred.html#E1:35_1_2_4_1_1_1"><i><font color="Green" title="E34">A30</font></i></a>, <a class="ref" href="polynom7.html#D7" title="POLYNOM7:def.7">POLYNOM7:def 7</a></i>;<br/><b>hence </b><a NAME="E4:35_1_2_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_4_1_1_2"><b>case </b></a><a NAME="E1:35_1_2_4_1_1_2"/>
<font color="Maroon" title="c10">u'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">u'</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="txt" href="polyred.html#E14"><i><font color="Green" title="E14">Lm8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">u'</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>
;<br/><b>hence </b><a NAME="E3:35_1_2_4_1_1_2"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:35_1_2_4"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 ;<br/></div><b>end;</b></div><a NAME="E7:35_1_2"/><b>then </b>
 <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="polyred.html#E5:35_1_2"><i><font color="Green" title="E33">A29</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E8:35_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="polyred.html#E4:35_1_2"><i><font color="Green" title="E32">A28</font></i></a>, <a class="ref" href="polyred.html#T5" target="_self" title="POLYRED:th.5">Th5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
