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<b>let </b><font color="Maroon" title="c6">F1</font>, <font color="Maroon" title="c7">F2</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:24_1_2"/><i><font color="Green" title="E20">A17</font></i>: 
( <font color="Maroon" title="c6">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> &amp; ( for <font color="Olive" title="b1">b</font>, <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>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F1</font> &amp; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b1">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F1</font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:24_1_2"/><i><font color="Green" title="E21">A18</font></i>: 
( <font color="Maroon" title="c7">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> &amp; ( for <font color="Olive" title="b1">b</font>, <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>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F2</font> &amp; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Olive" title="b1">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F2</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_2_1"/><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">u'</font> = <font color="Maroon" title="c8">u</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_1_1"/><i><font color="Green" title="E23">A20</font></i>: 
not <font color="Maroon" title="c9">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F2</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_2_1_1_1"/><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c10">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c9">u'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c11">x'</font>,<font color="Maroon" title="c2">T</font> or  not <font color="Maroon" title="c9">u'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c11">x'</font>,<font color="Maroon" title="c2">T</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:24_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">u'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c11">x'</font>,<font color="Maroon" title="c2">T</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F1</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="groeb_3.html#E2:24_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="groeb_3.html#E1:24_1_2_1_1_1"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="groeb_3.html#E1:24_1_2_1"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:24_1_2_1_1_1_1_1_2"/>
not <font color="Maroon" title="c9">u'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c11">x'</font>,<font color="Maroon" title="c2">T</font>
 ;<br/><div class="add"><a NAME="E2:24_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">x'</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c9">u'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#T5" title="TERMORD:th.5">TERMORD:5</a></i>;<br/><a NAME="E3:24_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">x'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c9">u'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></i>;<br/><b>hence </b><a NAME="E4:24_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F1</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_2_1_1"><i><font color="Green" title="E23">A20</font></i></a>, <a class="txt" href="groeb_3.html#E1:24_1_2_1_1_1"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="groeb_3.html#E2:24_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="groeb_3.html#E1:24_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="groeb_3.html#E1:24_1_2_1"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_2_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F1</font>
 ;<br/></div><b>end;</b></div><a NAME="E3:24_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c7">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E4:24_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c7">F2</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c6">F1</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_2_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="groeb_3.html#E1:24_1_2_1_1"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E5:24_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="groeb_3.html#E2:24_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_2_1"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F2</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:24_1_2"/><b>then </b>
<font color="Maroon" title="c6">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:24_1_2"/>
<font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:24_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="groeb_3.html#E2:24_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="triang_1.html#T3" title="TRIANG_1:th.3">TRIANG_1:3</a></i>;<br/>


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