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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  <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="pre_poly.html#NM2" title="PRE_POLY:NM.2">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> iff  <a href="termord.html#K2" title="TERMORD:func.2">max</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/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be   <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">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> iff  <a href="termord.html#K2" title="TERMORD:func.2">max</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/><span class="kw">let </span><font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">b2</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> iff  <a href="termord.html#K2" title="TERMORD:func.2">max</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><span class="lab"><font color="Green" title="E21">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 <a href="termord.html#K2" title="TERMORD:func.2">max</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_1_1"/>
(  not <font color="Maroon" title="c4">b2</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="termord.html#D5" target="_self" title="TERMORD:def.5">Def5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:30_1_1_1_1"/>
not <font color="Maroon" title="c4">b2</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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></span><br/><div class="add"><a NAME="E2:30_1_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="termord.html#E15"><span class="lab"><font color="Green" title="E13">Lm5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:30_1_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="c3">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D4" target="_self" title="TERMORD:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:30_1_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:30_1_1_1_2"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T11" target="_self" title="TERMORD:th.11">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:30_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="c3">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:30_2"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="termord.html#K2" title="TERMORD:func.2">max</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="30_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_2_1_1"/>
( <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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="termord.html#D4" target="_self" title="TERMORD:def.4">Def4</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:30_2_1_1_1"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="termord.html#K2" title="TERMORD:func.2">max</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"><a NAME="E2:30_2_1_1_1"/><span class="kw">then </span>
 <a href="termord.html#K2" title="TERMORD:func.2">max</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c3">b1</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 class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D5" target="_self" title="TERMORD:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:30_2_1_1_1"/>
 <a href="termord.html#K2" title="TERMORD:func.2">max</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 class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T15" target="_self" title="TERMORD:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:30_2_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="termord.html#K2" title="TERMORD:func.2">max</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"><span class="kw">hence </span><a NAME="E2:30_2_1_1_2"/>
 <a href="termord.html#K2" title="TERMORD:func.2">max</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 class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T12" target="_self" title="TERMORD:th.12">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:30_2"/>
 <a href="termord.html#K2" title="TERMORD:func.2">max</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:30"/>
(  <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> iff  <a href="termord.html#K2" title="TERMORD:func.2">max</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E21">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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