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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#V2" title="BAGORDER:attr.2">admissible</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>, <font color="Olive" title="b4">b3</font>, <font color="Olive" title="b5">b4</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b2">b1</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Olive" title="b3">b2</font>,<font color="Olive" title="b1">T</font> &amp; <font color="Olive" title="b4">b3</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b5">b4</font>,<font color="Olive" title="b1">T</font> holds <br/><font color="Olive" title="b2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Olive" title="b4">b3</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Olive" title="b3">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Olive" title="b5">b4</font>,<font color="Olive" title="b1">T</font></span><br/><b>let </b><font color="Maroon" title="c2">T</font> be  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</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:()"><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>, <font color="Olive" title="b3">b3</font>, <font color="Olive" title="b4">b4</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b1</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Olive" title="b2">b2</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b3">b3</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b4">b4</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b1">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Olive" title="b3">b3</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Olive" title="b2">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Olive" title="b4">b4</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">b2</font>, <font color="Maroon" title="c5">b3</font>, <font color="Maroon" title="c6">b4</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"> ( <font color="Maroon" title="c3">b1</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c5">b3</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c6">b4</font>,<font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font>,<font color="Maroon" title="c2">T</font> )</span><br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c3">b1</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c5">b3</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c6">b4</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"> <font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font>,<font color="Maroon" title="c2">T</font></span><br/>

<a NAME="E2:5"/><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> &amp; <font color="Maroon" title="c3">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font> )
 
<b>by </b><i><a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></i>;<br/>
<a NAME="E3:5"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">b3</font>,<font color="Maroon" title="c6">b4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> )
 
<b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="termord.html#D2" title="TERMORD:def.2">TERMORD:def 2</a></i>;<br/>
<a NAME="E4:5"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> )
 
<b>by </b><i><a class="ref" href="bagorder.html#D7" title="BAGORDER:def.7">BAGORDER:def 7</a></i>;<br/>
<a NAME="E5:5"/><b>then </b><i><font color="Green" title="E7">A2</font></i>: 
( <font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font>,<font color="Maroon" title="c2">T</font> )
 
<b>by </b><i><a class="ref" href="termord.html#D2" title="TERMORD:def.2">TERMORD:def 2</a></i>;<br/>
<a NAME="E6:5"/><b>then </b><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="termord.html#T8" title="TERMORD:th.8">TERMORD:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:5_1"/>
( <font color="Maroon" title="c6">b4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b4</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c4">b2</font> &amp; <font color="Maroon" title="c5">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b3</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c4">b2</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>;<br/><a NAME="E3:5_1"/><b>then </b><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font>
 <b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="#E1:5_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="termord.html#T7" title="TERMORD:th.7">TERMORD:7</a></i>;<br/><a NAME="E4:5_1"/>
( <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c6">b4</font> &amp; <font color="Maroon" title="c4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c6">b4</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>;<br/><b>hence </b><a NAME="E5:5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E3:5_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</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>
<b>hence </b><a NAME="E8:5"/>
<font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c5">b3</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c6">b4</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="#E6:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></i>; <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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