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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  holds <br/>(  <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font>,<font color="Olive" title="b1">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">b1</font> or  <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font>,<font color="Olive" title="b1">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">b2</font> )</span><br/><b>let </b><font color="Maroon" title="c2">T</font> be   <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  holds <br/>(  <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">b1</font> or  <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">b2</font> )</span><br/><b>let </b><font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">b2</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">b1</font> or  <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">b2</font> )</span><br/>





<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">b2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1"/>
(  not <font color="Maroon" title="c3">b1</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font> or <font color="Maroon" title="c3">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">b2</font> )
 </span><b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="termord.html#D4" target="_self" title="TERMORD:def.4">Def4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><b>case </b></a><a NAME="E1:24_1_1_1"/>
not <font color="Maroon" title="c3">b1</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">b2</font></span><br/><div class="add"><b>hence </b><a NAME="E2:24_1_1_1"/>
 <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">b2</font>
 <b>by </b><i><a class="ref" href="termord.html#D4" target="_self" title="TERMORD:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2"><b>case </b></a><a NAME="E1:24_1_1_2"/>
<font color="Maroon" title="c3">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:24_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">b1</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="termord.html#E11"><i><font color="Green" title="E9">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="termord.html#D4" target="_self" title="TERMORD:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
 <a href="termord.html#K1" title="TERMORD:func.1">min</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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