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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">b2</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>



<div><i><font color="Green" title="E5">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1"/><i><font color="Green" title="E6">A2</font></i>: 
not <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">b2</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1"/>
( <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1"><b>case </b></a><a NAME="E1:6_1_1_1_1"/>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_1_1"/>
<font color="Maroon" title="c3">b2</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c2">b1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2"><b>case </b></a><a NAME="E1:6_1_1_1_2"/>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b2</font>
 ;<br/><div class="add"><a NAME="E2:6_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c3">b2</font>
 <b>by </b><i><a class="txt" href="polyred.html#E1:6_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="polynom1.html#D12" title="POLYNOM1:def.12">POLYNOM1:def 12</a></i>;<br/><b>hence </b><a NAME="E3:6_1_1_1_2"/>
<font color="Maroon" title="c3">b2</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c2">b1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T49" title="POLYNOM1:th.49">POLYNOM1:49</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:6_1"/>
<font color="Maroon" title="c3">b2</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c2">b1</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c3">b2</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c2">b1</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_2_1_1"/>
( <font color="Maroon" title="c3">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b1</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_1"><b>case </b></a><a NAME="E1:6_2_1_1_1"/>
<font color="Maroon" title="c3">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_2_1_1_1"/>
not <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">b2</font>
 <b>by </b><i><a class="txt" href="polyred.html#E1:6_2"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="polynom1.html#D12" title="POLYNOM1:def.12">POLYNOM1:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_2"><b>case </b></a><a NAME="E1:6_2_1_1_2"/>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_2_1_1_2"/>
for <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <br/>(  not <font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> or  ex <font color="Olive" title="b2">l</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">k</font> &amp; not <font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">l</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:6_2"/>
not <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">b2</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:6"/>
( not <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">b2</font> iff <font color="Maroon" title="c3">b2</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c2">b1</font> )
 <b>by </b><i><a class="txt" href="polyred.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>


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