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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>] <b>means </b> ex <font color="Olive" title="b1">b1</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b2">b2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> ex <font color="Olive" title="b3">Q1</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b3">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b1">b1</font> &amp; $1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b2</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b2">b2</font> );<br/>
<a NAME="E1:14_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">b</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c6">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c7">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E2:14_1_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Q1</font> = <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c6">b1</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b2</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">b1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">b2</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">Q1</font>
;<br/>

<b>thus </b><a NAME="E4:14_1_1_1"/>
( <font color="Maroon" title="c8">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c5">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c7">b2</font> &amp; <font color="Maroon" title="c8">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b2</font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E2:14_1_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E3:14_1_1"/><i><font color="Green" title="E10">A3</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c5">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b1">b</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E1:14_1_1"><i><font color="Green" title="E9">A1</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</font> as   <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default">$1 <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font></span>)</span> where <font color="Olive" title="b1">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> :  ex <font color="Olive" title="b2">Q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> $1 &amp; <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Olive" title="b2">Q</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">A</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b1</font>) where <font color="Olive" title="b1">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span> ;<br/>
<a NAME="E5:14_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></i>;<br/>
<a NAME="E6:14_1_1"/><i><font color="Green" title="E12">A5</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b1</font>) where <font color="Olive" title="b1">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="polynom6.html#E5:14_1_1"><i><font color="Green" title="E11">A4</font></i></a>);<br/></i>
<a NAME="E7:14_1_1"/>
for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b1</font>) where <font color="Olive" title="b2">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b2">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span>  holds <br/><font color="Olive" title="b1">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_1_1_2"/>
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b1</font>) where <font color="Olive" title="b1">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font><b> such that </b><br/><a NAME="E3:14_1_1_2"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font></span>)</span> where <font color="Olive" title="b1">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> :  ex <font color="Olive" title="b2">Q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c9">b1</font> &amp; <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Olive" title="b2">Q</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:14_1_1_2"/>
<font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">Q0</font> = <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c9">b1</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font>) <b>-&gt; </b>   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> = <font color="Maroon" title="c9">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> $1;<br/>
<b>set </b><font color="Maroon" title="c11">B0</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b2</font>) where <font color="Olive" title="b1">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> : <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c10">Q0</font> </span>}</span> ;<br/>
<a NAME="E6:14_1_1_2"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b2</font>) where <font color="Olive" title="b1">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> : <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c10">Q0</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:14_1_1_2_1"/>
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b2</font>) where <font color="Olive" title="b1">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> : <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c10">Q0</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_2_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E3:14_1_1_2_1_1"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c13">b2</font>
 <b>and </b><br/><a NAME="E4:14_1_1_2_1_1"/><i><font color="Green" title="E16">A9</font></i>: 
 ex <font color="Olive" title="b1">Q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c9">b1</font> &amp; <font color="Maroon" title="c13">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Olive" title="b1">Q</font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E3:14_1_1_2"><i><font color="Green" title="E13">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">Q</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E6:14_1_1_2_1_1"/><i><font color="Green" title="E17">A10</font></i>: 
<font color="Maroon" title="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c9">b1</font>
 <b>and </b><br/><a NAME="E7:14_1_1_2_1_1"/><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c13">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c14">Q</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:14_1_1_2_1_1"><i><font color="Green" title="E16">A9</font></i></a></i>;<br/>
<b>thus </b><a NAME="E8:14_1_1_2_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b2</font>) where <font color="Olive" title="b1">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> : <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c10">Q0</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="polynom6.html#E3:14_1_1_2_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="polynom6.html#E6:14_1_1_2_1_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="polynom6.html#E7:14_1_1_2_1_1"><i><font color="Green" title="E18">A11</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:14_1_1_2_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b2</font>) where <font color="Olive" title="b1">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> : <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c10">Q0</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_2_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b2</font>) where <font color="Olive" title="b1">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> : <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c10">Q0</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E3:14_1_1_2_1_2"/><i><font color="Green" title="E15">A12</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c13">b2</font>
 <b>and </b><br/><a NAME="E4:14_1_1_2_1_2"/><i><font color="Green" title="E16">A13</font></i>: 
<font color="Maroon" title="c13">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c10">Q0</font>
 ;<br/>
<b>thus </b><a NAME="E5:14_1_1_2_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:14_1_1_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="polynom6.html#E3:14_1_1_2_1_2"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="polynom6.html#E4:14_1_1_2_1_2"><i><font color="Green" title="E16">A13</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E7:14_1_1_2"/><i><font color="Green" title="E15">A14</font></i>: 
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c10">Q0</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<a NAME="E8:14_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b2</font>) where <font color="Olive" title="b1">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> : <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c10">Q0</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="polynom6.html#E7:14_1_1_2"><i><font color="Green" title="E15">A14</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:14_1_1_2"/>
<font color="Maroon" title="c8">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="polynom6.html#E6:14_1_1_2"><i><font color="Green" title="E14">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:14_1_1"/><b>then </b><i><font color="Green" title="E13">A15</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b1</font>) where <font color="Olive" title="b1">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:14_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a></i>;<br/>
<a NAME="E9:14_1_1"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b1</font>) where <font color="Olive" title="b1">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:14_1_1_3"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b1</font>) where <font color="Olive" title="b1">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_3_1"/><i><font color="Green" title="E14">A16</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c6">f</font>
 ;<br/>

<a NAME="E2:14_1_1_3_1"/><b>then </b><i><font color="Green" title="E15">A17</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</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="c3">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c10">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E4:14_1_1_3_1"/><i><font color="Green" title="E16">A18</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c10">b2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:14_1_1_3_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:14_1_1_3_1"/><i><font color="Green" title="E17">A19</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2'</font></span>)</span> where <font color="Olive" title="b1">b2'</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> :  ex <font color="Olive" title="b2">Q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c9">b1</font> &amp; <font color="Olive" title="b1">b2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Olive" title="b2">Q</font> ) </span>}</span> 
 ;<br/>
<b>consider </b><font color="Maroon" title="c12">b1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c13">b2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c14">Q1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E8:14_1_1_3_1"/><i><font color="Green" title="E18">A20</font></i>: 
<font color="Maroon" title="c14">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c12">b1'</font>
 <b>and </b><br/><a NAME="E9:14_1_1_3_1"/><i><font color="Green" title="E19">A21</font></i>: 
<font color="Maroon" title="c9">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c10">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c12">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c13">b2'</font>
 <b>and </b><br/><a NAME="E10:14_1_1_3_1"/><i><font color="Green" title="E20">A22</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c13">b2'</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:14_1_1"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>
<a NAME="E11:14_1_1_3_1"/><i><font color="Green" title="E21">A23</font></i>: 
( <font color="Maroon" title="c9">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c12">b1'</font> &amp; <font color="Maroon" title="c10">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">b2'</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E9:14_1_1_3_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<a NAME="E12:14_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c10">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c14">Q1</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E2:14_1_1_3_1"><i><font color="Green" title="E15">A17</font></i></a>, <a class="txt" href="polynom6.html#E4:14_1_1_3_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="polynom6.html#E10:14_1_1_3_1"><i><font color="Green" title="E20">A22</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E13:14_1_1_3_1"/><b>then </b><i><font color="Green" title="E22">A24</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:14_1_1_3_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="polynom6.html#E6:14_1_1_3_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="polynom6.html#E8:14_1_1_3_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="polynom6.html#E11:14_1_1_3_1"><i><font color="Green" title="E21">A23</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_1_3_1_1"/>
<font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 ;<br/><a NAME="E2:14_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c14">Q1</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="c2">o2</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:14_1_1_3_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="polynom6.html#E11:14_1_1_3_1"><i><font color="Green" title="E21">A23</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>hence </b><a NAME="E3:14_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom6.html#E2:14_1_1_3_1"><i><font color="Green" title="E15">A17</font></i></a>, <a class="txt" href="polynom6.html#E4:14_1_1_3_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="polynom6.html#E10:14_1_1_3_1"><i><font color="Green" title="E20">A22</font></i></a>, <a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:14_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E16:14_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b1</font>) where <font color="Olive" title="b1">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:14_1_1_3_1"><i><font color="Green" title="E17">A19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:14_1_1_3_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b1</font>) where <font color="Olive" title="b1">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="polynom6.html#E13:14_1_1_3_1"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:14_1_1_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b1</font>) where <font color="Olive" title="b1">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:14_1_1_3"/><i><font color="Green" title="E14">A25</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font>
 <b>and </b><br/><a NAME="E5:14_1_1_3"/><i><font color="Green" title="E15">A26</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b1</font>) where <font color="Olive" title="b1">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> : <font color="Olive" title="b1">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font><b> such that </b><br/><a NAME="E7:14_1_1_3"/><i><font color="Green" title="E16">A27</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font></span>)</span> where <font color="Olive" title="b1">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> :  ex <font color="Olive" title="b2">Q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">b1</font> &amp; <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Olive" title="b2">Q</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E8:14_1_1_3"/>
<font color="Maroon" title="c10">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:14_1_1_3"><i><font color="Green" title="E15">A26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E10:14_1_1_3"/><i><font color="Green" title="E17">A28</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c11">b2</font>
 <b>and </b><br/><a NAME="E11:14_1_1_3"/><i><font color="Green" title="E18">A29</font></i>: 
 ex <font color="Olive" title="b1">Q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c11">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Olive" title="b1">Q</font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E4:14_1_1_3"><i><font color="Green" title="E14">A25</font></i></a>, <a class="txt" href="polynom6.html#E7:14_1_1_3"><i><font color="Green" title="E16">A27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">Q</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E13:14_1_1_3"/><i><font color="Green" title="E19">A30</font></i>: 
<font color="Maroon" title="c12">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">b1</font>
 <b>and </b><br/><a NAME="E14:14_1_1_3"/><i><font color="Green" title="E20">A31</font></i>: 
<font color="Maroon" title="c11">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E11:14_1_1_3"><i><font color="Green" title="E18">A29</font></i></a></i>;<br/>
<a NAME="E15:14_1_1_3"/><i><font color="Green" title="E21">A32</font></i>: 
<font color="Maroon" title="c12">Q</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">b2</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="c3">L</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E14:14_1_1_3"><i><font color="Green" title="E20">A31</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">b1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c14">b2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c15">Q1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E17:14_1_1_3"/><i><font color="Green" title="E22">A33</font></i>: 
<font color="Maroon" title="c15">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c13">b1'</font>
 <b>and </b><br/><a NAME="E18:14_1_1_3"/><i><font color="Green" title="E23">A34</font></i>: 
<font color="Maroon" title="c10">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c11">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c14">b2'</font>
 <b>and </b><br/><a NAME="E19:14_1_1_3"/><i><font color="Green" title="E24">A35</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c11">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c14">b2'</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:14_1_1"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>
<a NAME="E20:14_1_1_3"/><i><font color="Green" title="E25">A36</font></i>: 
<font color="Maroon" title="c15">Q1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">Q</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E13:14_1_1_3"><i><font color="Green" title="E19">A30</font></i></a>, <a class="txt" href="polynom6.html#E17:14_1_1_3"><i><font color="Green" title="E22">A33</font></i></a>, <a class="txt" href="polynom6.html#E18:14_1_1_3"><i><font color="Green" title="E23">A34</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<a NAME="E21:14_1_1_3"/>
<font color="Maroon" title="c6">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c11">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">b2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E18:14_1_1_3"><i><font color="Green" title="E23">A34</font></i></a>, <a class="txt" href="polynom6.html#E19:14_1_1_3"><i><font color="Green" title="E24">A35</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E22:14_1_1_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E10:14_1_1_3"><i><font color="Green" title="E17">A28</font></i></a>, <a class="txt" href="polynom6.html#E15:14_1_1_3"><i><font color="Green" title="E21">A32</font></i></a>, <a class="txt" href="polynom6.html#E20:14_1_1_3"><i><font color="Green" title="E25">A36</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c6">f</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="txt" href="polynom6.html#E8:14_1_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c9">b</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c10">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c11">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E12:14_1_1"/><i><font color="Green" title="E14">A37</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c10">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c11">b2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">b1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c11">b2</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c12">Q1</font> = <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">b1</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">Q1</font>
;<br/>

<b>thus </b><a NAME="E14:14_1_1"/>
<font color="Maroon" title="c12">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c10">b1</font>
 ;<br/>

<b>thus </b><a NAME="E15:14_1_1"/>
<font color="Maroon" title="c9">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c10">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c11">b2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E12:14_1_1"><i><font color="Green" title="E14">A37</font></i></a></i>;<br/>

<b>consider </b><font color="Maroon" title="c13">b1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c14">b2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c15">Q1'</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E17:14_1_1"/><i><font color="Green" title="E15">A38</font></i>: 
<font color="Maroon" title="c15">Q1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c13">b1'</font>
 <b>and </b><br/><a NAME="E18:14_1_1"/><i><font color="Green" title="E16">A39</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c14">b2'</font>
 <b>and </b><br/><a NAME="E19:14_1_1"/><i><font color="Green" title="E17">A40</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Q1'</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c14">b2'</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:14_1_1"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>
<a NAME="E20:14_1_1"/>
( <font color="Maroon" title="c10">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">b1'</font> &amp; <font color="Maroon" title="c11">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b2'</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E12:14_1_1"><i><font color="Green" title="E14">A37</font></i></a>, <a class="txt" href="polynom6.html#E18:14_1_1"><i><font color="Green" title="E16">A39</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E21:14_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">b2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E17:14_1_1"><i><font color="Green" title="E15">A38</font></i></a>, <a class="txt" href="polynom6.html#E19:14_1_1"><i><font color="Green" title="E17">A40</font></i></a></i>;<br/>


</div>
