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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>set </b><font color="Maroon">CUT1</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"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E1:26"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</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> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span> or  not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:26_1_1_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">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">x</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">x</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="E4:26_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</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>, </i>;<br/><b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><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"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</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">k'</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>
;<br/><b>hence </b><a NAME="E7:26_1_1_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"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_2"/>
not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:26_1_1_2"/><b>then </b>
<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"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_2"/>
<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"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:26"/><b>then </b>
<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"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#R3">divides</a>  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><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="polynom1.html#D13">POLYNOM1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:26"/>
<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"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><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="polynom1.html#T62">POLYNOM1:62</a></i>;<br/>


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