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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">f</font> being <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-support</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-support</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <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">  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c3">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">X</font> )</span><br/>





<b>assume </b><a NAME="E1:123"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="polynom1.html#K16" title="POLYNOM1:func.16">NatMinor</a> <font color="Maroon" title="c2">f</font> ;<br/>
<a NAME="E3:123"/><i><font color="Green" title="E71">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/>
<a NAME="E4:123"/><b>then </b><i><font color="Green" title="E72">A2</font></i>: 
<font color="Maroon" title="c4">x'</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:123"/>
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">x'</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<b>assume </b><a NAME="E1:123_1"/><i><font color="Green" title="E73">A3</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">x'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">f</font></span><br/>

<a NAME="E2:123_1"/><b>then </b>
<font color="Maroon" title="c4">x'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self" title="POLYNOM1:def.7">Def7</a></i>;<br/>
<a NAME="E3:123_1"/><b>then </b><i><font color="Green" title="E74">A4</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
 
;<br/>
<a NAME="E4:123_1"/>
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">x'</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41" target="_self" title="POLYNOM1:th.41">Th41</a></i>;<br/>
<a NAME="E5:123_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E3:123"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="#E4:123"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="#E1:123_1"><i><font color="Green" title="E73">A3</font></i></a>, <a class="txt" href="#E3:123_1"><i><font color="Green" title="E74">A4</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self" title="POLYNOM1:def.17">Def17</a></i>;<br/>
<b>hence </b><a NAME="E6:123_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self" title="POLYNOM1:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:123"/><b>then </b>
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <font color="Maroon" title="c4">x'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<a NAME="E7:123"/><b>then </b>
<font color="Maroon" title="c3">x</font> is   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E3:123"><i><font color="Green" title="E71">A1</font></i></a>, <a class="ref" href="polynom1.html#D8" target="_self" title="POLYNOM1:def.8">Def8</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:123"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</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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