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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">  <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font></span><br/>

<b>set </b><font color="Maroon" title="c2">BO</font> =  <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E1:117"/><i><font color="Green" title="E68">A1</font></i>: 
 <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E81"><i><font color="Green" title="E64">Lm1</font></i></a></i>;<br/>
<a NAME="E2:117"/><i><font color="Green" title="E69">A2</font></i>: 
 <a href="polynom1.html#K15" title="POLYNOM1:func.15">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="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E82"><i><font color="Green" title="E65">Lm2</font></i></a></i>;<br/>
<a NAME="E3:117"/><i><font color="Green" title="E70">A3</font></i>: 
 <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E83"><i><font color="Green" title="E66">Lm3</font></i></a></i>;<br/>
<a NAME="E4:117"/>
 <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>proof </b></a><div class="add">

<b>let </b><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="polynom1.html#K12" title="POLYNOM1:func.12">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="polynom1.html#K12" title="POLYNOM1:func.12">Bags</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> or <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="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> or <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="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> )</span><br/>

<b>assume </b><a NAME="E1:117_1"/><i><font color="Green" title="E71">A4</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">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="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; 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="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <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"> <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="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c4">y</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/>
<a NAME="E3:117_1"/>
not <font color="Maroon" title="c5">p</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c6">q</font>
 
<b>by </b><i><a class="txt" href="#E1:117_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="polynom1.html#D16" target="_self" title="POLYNOM1:def.16">Def16</a></i>;<br/>
<a NAME="E4:117_1"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T49" target="_self" title="POLYNOM1:th.49">Th49</a></i>;<br/>
<b>hence </b><a NAME="E5:117_1"/>
<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="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D16" target="_self" title="POLYNOM1:def.16">Def16</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="E5:117"/>
 <a href="polynom1.html#K15" title="POLYNOM1:func.15">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="#E1:117"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="#E2:117"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="#E3:117"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="orders_1.html#D8" title="ORDERS_1:def.8">ORDERS_1:def 8</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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