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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">  <a href="pre_poly.html#K18" title="PRE_POLY:func.18">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">BO</font> =  <a href="pre_poly.html#K18" title="PRE_POLY:func.18">BagOrder</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> or  not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K18" title="PRE_POLY:func.18">BagOrder</a> <font color="Maroon" title="c1">n</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K18" title="PRE_POLY:func.18">BagOrder</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:133"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">and </span><br/><a NAME="E2:133"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K18" title="PRE_POLY:func.18">BagOrder</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E3:133"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K18" title="PRE_POLY:func.18">BagOrder</a> <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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">b1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">b2</font> = <font color="Maroon" title="c4">y</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="pre_poly.html#D12" target="_self" title="PRE_POLY:def.12">Def14</a></span>;<br/>
<a NAME="E5:133"/>
<font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c6">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="pre_poly.html#D14" target="_self" title="PRE_POLY:def.14">Def16</a></span>;<br/>
<a NAME="E6:133"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A4</font></span>: 
( <font color="Maroon" title="c5">b1</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c6">b2</font> or <font color="Maroon" title="c5">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">b2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D10" target="_self" title="PRE_POLY:def.10">Def12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">b19</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c8">b29</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="pre_poly.html#D12" target="_self" title="PRE_POLY:def.12">Def14</a></span>;<br/>
<a NAME="E8:133"/>
<font color="Maroon" title="c7">b19</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c8">b29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:133"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="pre_poly.html#D14" target="_self" title="PRE_POLY:def.14">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:133"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:133"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="pre_poly.html#D10" target="_self" title="PRE_POLY:def.10">Def12</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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