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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> ;<br/><b>let </b><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">T</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 <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:15"/>
<font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">T</font>
 ;<br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
(  not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> )
 </span><b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1"/><i><font color="Green" title="E14">A2</font></i>: 
not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1"/><b>then </b><i><font color="Green" title="E15">A3</font></i>: 
( not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font> = 
0
<b>by </b><i><a class="txt" href="uproots.html#E1:15_1_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="uproots.html#T8" target="_self" title="UPROOTS:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0
<b>by </b><i><a class="txt" href="uproots.html#E2:15_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="uproots.html#T8" target="_self" title="UPROOTS:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E2:15_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="uproots.html#T8" target="_self" title="UPROOTS:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2"/><b>then </b><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:15_1_1_2"/><i><font color="Green" title="E16">A6</font></i>: 
not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="uproots.html#E2:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="uproots.html#E1:15_1_1_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="txt" href="uproots.html#E2:15_1_1_2"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="uproots.html#T9" target="_self" title="UPROOTS:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0
<b>by </b><i><a class="txt" href="uproots.html#E1:15_1_1_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="uproots.html#T9" target="_self" title="UPROOTS:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E3:15_1_1_2"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="uproots.html#T8" target="_self" title="UPROOTS:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_3"><b>suppose </b></a><a NAME="E1:15_1_1_3"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_3"/><b>then </b><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:15_1_1_3"/><i><font color="Green" title="E16">A9</font></i>: 
not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="uproots.html#E2:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="uproots.html#E1:15_1_1_3"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="txt" href="uproots.html#E2:15_1_1_3"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="uproots.html#T9" target="_self" title="UPROOTS:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0
<b>by </b><i><a class="txt" href="uproots.html#E1:15_1_1_3"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="uproots.html#T9" target="_self" title="UPROOTS:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E3:15_1_1_3"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="uproots.html#T8" target="_self" title="UPROOTS:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font> <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


</div>
