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<div class="add">

<b>let </b><font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c2">o2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">b1</font>, <font color="Maroon" title="c5">c1</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>;<br/><b>let </b><font color="Maroon" title="c6">a2</font>, <font color="Maroon" title="c7">b2</font>, <font color="Maroon" title="c8">c2</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>;<br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c5">c1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c8">c2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">b2</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c6">a2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">w</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">a2</font></span>)</span> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E3:19"/>
for <font color="Olive" title="b1">o</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds <br/> ( ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o1</font> implies <font color="Maroon" title="c9">w</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">o</font> ) &amp; ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">o1</font> implies <font color="Maroon" title="c9">w</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c1">o1</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">o</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1_1"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o1</font>
 ;<br/><b>thus </b><font color="Maroon" title="c9">w</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">a2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">a2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E1:19_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self" title="POLYNOM6:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E1:19_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self" title="POLYNOM6:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c5">c1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font>
<b>by </b><i><a class="txt" href="polynom6.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></i>
;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:19_1"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">o1</font>
 ;<br/>

<b>thus </b><font color="Maroon" title="c9">w</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">a2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">o</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c1">o1</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">a2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E2:19_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self" title="POLYNOM6:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">o</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c1">o1</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">o</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c1">o1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E2:19_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self" title="POLYNOM6:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c8">c2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">o</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c1">o1</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">a2</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c8">c2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self" title="POLYNOM6:def.1">Def1</a></i>;<br/>


</div>
