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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><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#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/><b>let </b><font color="Maroon" title="c3">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>set </b><font color="Maroon" title="c4">R</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #);<br/>
<b>set </b><font color="Maroon" title="c5">m</font> =  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <font color="Maroon" title="c3">P</font>,<a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #);<br/>
<a NAME="E1:17"/><i><font color="Green" title="E9">A1</font></i>: 
(  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <font color="Maroon" title="c3">P</font>,<a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <font color="Maroon" title="c3">P</font>,<a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #) <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c3">P</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #) )
 
<b>by </b><i><a class="ref" href="bagorder.html#D19" title="BAGORDER:def.19">BAGORDER:def 19</a></i>;<br/>
<a NAME="E2:17"/>
 <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <font color="Maroon" title="c3">P</font>,<a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #) is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">b</font> =  <a href="bagorder.html#K15" title="BAGORDER:func.15">MinElement</a> <font color="Maroon" title="c3">P</font>,<a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #) as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/>
<a NAME="E4:17"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c2">T</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">b'</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>assume </b><a NAME="E1:17_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c7">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ;<br/><a NAME="E2:17_1"/>
( <font color="Maroon" title="c6">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c7">b'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c7">b'</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="termord.html#T6" title="TERMORD:th.6">TERMORD:6</a></i>;<br/><a NAME="E3:17_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c6">b</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="c7">b'</font>,<font color="Maroon" title="c7">b'</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="ref" href="termord.html#D2" title="TERMORD:def.2">TERMORD:def 2</a></i>;<br/><a NAME="E4:17_1"/><b>then </b><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c7">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1"/>
( <font color="Maroon" title="c7">b'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c7">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1"><b>case </b></a><a NAME="E1:17_1_1_1_1"/>
<font color="Maroon" title="c7">b'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c7">b'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#T6" title="TERMORD:th.6">TERMORD:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_2"><b>case </b></a><a NAME="E1:17_1_1_1_2"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c7">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_2"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">b'</font>,<font color="Maroon" title="c6">b</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="groeb_2.html#E1:17"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="groeb_2.html#E1:17_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="waybel_4.html#D26" title="WAYBEL_4:def.26">WAYBEL_4:def 26</a></i>;<br/><a NAME="E3:17_1_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b'</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="groeb_2.html#E4:17"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="groeb_2.html#E4:17_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="groeb_2.html#E1:17_1_1_1_2"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_1_2"/>
<font color="Maroon" title="c6">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c7">b'</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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:17_1"/>
<font color="Maroon" title="c6">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c7">b'</font>,<font color="Maroon" title="c2">T</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:17"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; ( for <font color="Olive" title="b2">b'</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">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/><font color="Olive" title="b1">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c2">T</font> ) )
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:17"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


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