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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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/> for <font color="Olive" title="b2">p</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b3">m</font> being  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b4">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">m</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b3">m</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="Olive" title="b2">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b4">b</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">m</font> being  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b3">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b2">m</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="Olive" title="b1">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b3">b</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</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">m</font> being  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Olive" title="b1">m</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> <font color="Olive" title="b2">b</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">m</font> be   <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</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">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</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> <font color="Olive" title="b1">b</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">b2</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</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> <font color="Maroon" title="c5">b2</font></span>)</span></span><br/>









<b>set </b><font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c4">m</font> <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="c7">b</font> = <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font>;<br/>
<b>consider </b><font color="Maroon" title="c8">s</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="E2:18"/><i><font color="Green" title="E16">A1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span> <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="c8">s</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></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="c8">s</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></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="c8">s</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"><font color="Maroon" title="c4">m</font> <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/>
<i><font color="Green" title="E17">A2</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</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> <font color="Maroon" title="c8">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/>.= 
 <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E2:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c9">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> <b> such that </b><br/><a NAME="E5:18"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c5">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T73" title="POLYNOM1:th.73">POLYNOM1:73</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">b1'</font>, <font color="Maroon" title="c11">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="E7:18"/><i><font color="Green" title="E19">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c10">b1'</font>,<font color="Maroon" title="c11">b2'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c8">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">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="c11">b2'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polyred.html#E2:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="polyred.html#E3:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="polyred.html#E5:18"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/>
<i><font color="Green" title="E20">A5</font></i>:  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c10">b1'</font>,<font color="Maroon" title="c11">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#E5:18"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="polyred.html#E7:18"><i><font color="Green" title="E19">A4</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="c10">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/>
<i><font color="Green" title="E21">A6</font></i>: <font color="Maroon" title="c5">b2</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c5">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> 2
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c11">b2'</font>
<b>by </b><i><a class="txt" href="polyred.html#E5:18"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="polyred.html#E7:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E10:18"/>
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="c8">s</font> &amp; <font color="Olive" title="b1">k'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font> holds <br/><font color="Maroon" title="c8">s</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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">k'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">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="c8">s</font> &amp; <font color="Maroon" title="c12">k'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font> implies <font color="Maroon" title="c8">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">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> )</span><br/>

<b>assume </b><a NAME="E1:18_4"/><i><font color="Green" title="E22">A7</font></i>: 
( <font color="Maroon" title="c12">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="c8">s</font> &amp; <font color="Maroon" title="c12">k'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">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></span><br/>

<b>then consider </b><font color="Maroon" title="c13">b1'</font>, <font color="Maroon" title="c14">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="E3:18_4"/><i><font color="Green" title="E23">A8</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">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="c13">b1'</font>,<font color="Maroon" title="c14">b2'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c8">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c13">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="c14">b2'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polyred.html#E2:18"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>
<a NAME="E4:18_4"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c13">b1'</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k'</font>
 
<b>by </b><i><a class="txt" href="polyred.html#E3:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="polyred.html#E1:18_4"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="polyred.html#E3:18_4"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="polynom1.html#T74" title="POLYNOM1:th.74">POLYNOM1:74</a></i>;<br/>
<i><font color="Green" title="E25">A10</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c13">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="c13">b1'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c13">b1'</font></span>)</span></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="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c13">b1'</font>,<font color="Maroon" title="c14">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> 2
<b>by </b><i><a class="txt" href="polyred.html#E3:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="polyred.html#E1:18_4"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="polyred.html#E3:18_4"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="polyred.html#E4:18_4"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>
<br/>.= 
<font color="Maroon" title="c14">b2'</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="hsNdiv(this)" href="javascript:()" title="18_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_4_2"/>
( ( <font color="Maroon" title="c13">b1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c14">b2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> ) or <font color="Maroon" title="c13">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> <font color="Maroon" title="c4">m</font> or <font color="Maroon" title="c14">b2'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_4_2_1"><b>suppose </b></a><a NAME="E1:18_4_2_1"/><i><font color="Green" title="E26">A11</font></i>: 
( <font color="Maroon" title="c13">b1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c14">b2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c8">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">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></span><br/><div class="add"><span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k'</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k'</font>
<b>by </b><i><a class="txt" href="polyred.html#E3:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="polyred.html#E1:18_4"><i><font color="Green" title="E22">A7</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="txt" href="polyred.html#E5:18"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="polyred.html#E3:18_4"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="polyred.html#E1:18_4_2_1"><i><font color="Green" title="E26">A11</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="E3:18_4_2_1"/>
<font color="Maroon" title="c8">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">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#E3:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="polyred.html#E5:18"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="polyred.html#E1:18_4"><i><font color="Green" title="E22">A7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_4_2_2"><b>suppose </b></a><a NAME="E1:18_4_2_2"/>
<font color="Maroon" title="c13">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> <font color="Maroon" title="c4">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">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></span><br/><div class="add"><a NAME="E2:18_4_2_2"/><b>then </b>
<font color="Maroon" title="c4">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c13">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/><b>hence </b><a NAME="E3:18_4_2_2"/>
<font color="Maroon" title="c8">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">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#E3:18_4"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</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_4_2_3"><b>suppose </b></a><a NAME="E1:18_4_2_3"/>
<font color="Maroon" title="c14">b2'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">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="c8">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">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></span><br/><div class="add"><a NAME="E2:18_4_2_3"/><b>then </b>
<font color="Maroon" title="c13">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> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="polyred.html#E5:18_4"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>;<br/><a NAME="E3:18_4_2_3"/><b>then </b>
<font color="Maroon" title="c4">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c13">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/><b>hence </b><a NAME="E4:18_4_2_3"/>
<font color="Maroon" title="c8">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">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#E3:18_4"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</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>
<b>hence </b><a NAME="E11:18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m</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> <font color="Maroon" title="c5">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polyred.html#E2:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="polyred.html#E3:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="polyred.html#E5:18"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="polyred.html#E7:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="polyred.html#E8:18"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="polyred.html#E9:18"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="polynom2.html#T5" title="POLYNOM2:th.5">POLYNOM2:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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