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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">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>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c3">n</font>;<br/>



<b>set </b><font color="Maroon" title="c6">CUTAB</font> = <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c5">b</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">CUTA</font> = <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font>;<br/>
<b>set </b><font color="Maroon" title="c8">CUTB</font> = <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">i</font>
 ;<br/><a NAME="E2:27_1"/>
<font color="Maroon" title="c2">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">i</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="axioms.html#T30" title="AXIOMS:th.30">AXIOMS:30</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">k</font> being    <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> <b> such that </b><br/><a NAME="E4:27_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">i</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:27_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c9">x</font> as    <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>  <b>by </b><i><a class="txt" href="bagorder.html#E4:27_1"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/><a NAME="E6:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">x'</font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:27_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="bagorder.html#D1" target="_self" title="BAGORDER:def.1">Def1</a></i>;<br/><a NAME="E7:27_1"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">x'</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">x'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>;<br/><a NAME="E8:27_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">x'</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">x'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:27_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="bagorder.html#D1" target="_self" title="BAGORDER:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E9:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">x</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="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:27_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27"/>
<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c5">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


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