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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>



<b>set </b><font color="Maroon">CUTAB</font> = <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">CUTA</font> = <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">CUTB</font> = <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E2:27_1"/>
<font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:27_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E6:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x'</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:27_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x'</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>;<br/><a NAME="E8:27_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x'</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x'</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27"/>
<font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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