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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>,   <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>) <b>-&gt; </b>   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">n</font> = <font color="Maroon">a<sub>1</sub></font> <a href="polynom1.html#K9" target="_self">+</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>,<font color="Olive">b2</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <font color="Olive">b2</font> is    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> : ( <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">p</font> &amp; <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">q</font> ) </span>}</span> ;<br/>
<a NAME="E1:186_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">p</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:186_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">q</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:186_1_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>,<font color="Olive">b2</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <font color="Olive">b2</font> is    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> : ( <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">p</font> &amp; <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">q</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S22">FRAENKEL:sch 22</a>(<a class="ref" href="polynom1.html#T30" target="_self">Th30</a>, <a class="ref" href="polynom1.html#T31" target="_self">Th31</a>);<br/></i>
<a NAME="E4:186_1_1"/>
 <a href="polynom1.html#K12" target="_self">Support</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>,<font color="Olive">b2</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <font color="Olive">b2</font> is    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> : ( <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">p</font> &amp; <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">q</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="186_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x'</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:186_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">x'</font> as    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> ;<br/>
<a NAME="E3:186_1_1_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T32" target="_self">Th32</a>, <a class="txt" href="polynom1.html#E5:186_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:186_1_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E6:186_1_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E7:186_1_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T24" target="_self">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:186_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E10:186_1_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="txt" href="polynom1.html#E13:186_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="matrlin.html#T15">MATRLIN:15</a></i>;<br/>
<b>consider </b><font color="Maroon">b1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E12:186_1_1_1"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E13:186_1_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom1.html#E16:186_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="polynom1.html#E19:186_1_1_1"><i><font color="Green">E111</font></i></a></i>;<br/>
<a NAME="E14:186_1_1_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E15:186_1_1_1"/>
( <font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> &amp; <font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="txt" href="polynom1.html#E20:186_1_1_1"><i><font color="Green">E112</font></i></a>, , <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>;<br/>
<a NAME="E16:186_1_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="txt" href="polynom1.html#E5:186_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E17:186_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E14:186_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="polynom1.html#E19:186_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then consider </b><font color="Maroon">b1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E19:186_1_1_1"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1'</font>,<font color="Maroon">b2'</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E20:186_1_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> <a href="polynom1.html#K9" target="_self">+</a> <font color="Maroon">b2'</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E21:186_1_1_1"/>
( <font color="Maroon">b1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2</font> )
 
<b>by </b><i>, , <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<b>hence </b><a NAME="E22:186_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>,<font color="Olive">b2</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <font color="Olive">b2</font> is    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> : ( <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">p</font> &amp; <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12" target="_self">Support</a> <font color="Maroon">q</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="polynom1.html#T33" target="_self">Th33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:186_1_1"/><b>then </b>
 <a href="polynom1.html#K12" target="_self">Support</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>hence </b><a NAME="E6:186_1_1"/>
<font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font> is <a href="polynom1.html#V2" target="_self">finite-Support</a>
 <b>by </b><i/>;<br/>


</div>
