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<span class="kw">let </span><font color="Maroon" title="c1">o</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">gilo</font> =  <a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">ilo</font> =  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font>;<br/>
<a NAME="E1:49"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 <a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#D5" target="_self" onmouseover="rs('bagorder/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E2:49"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T12" onmouseover="rs('orders_1/T12')" onmouseout="rh()">ORDERS_1:12</a></span>;<br/>
<a NAME="E3:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 <a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D9" onmouseover="rs('relat_2/D9')" onmouseout="rh()">RELAT_2:def 9</a></span>;<br/>
<a NAME="E4:49"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 <a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="relat_2.html#D16" onmouseover="rs('relat_2/D16')" onmouseout="rh()">RELAT_2:def 16</a></span>;<br/>
<a NAME="E5:49"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
 <a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="relat_2.html#D12" onmouseover="rs('relat_2/D12')" onmouseout="rh()">RELAT_2:def 12</a></span>;<br/>
<a NAME="E6:49"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>;<br/>
<a NAME="E7:49"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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>  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Olive" title="b3">c</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>  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#D5" target="_self" onmouseover="rs('bagorder/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> &amp; <font color="Olive" title="b1">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b2">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b2">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Y</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">Y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> &amp; <font color="Maroon" title="c4">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49_1"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:49_1"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c5">TD</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> </span> } </span> ;<br/><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:49_1"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="orders_1.html#T12" onmouseover="rs('orders_1/T12')" onmouseout="rh()">ORDERS_1:12</a></span>;<br/><a NAME="E6:49_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a></span>;<br/><a NAME="E7:49_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> </span> } </span>  or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> </span><br/>

<a NAME="E2:49_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:49_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">TD</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> </span> } </span>  as   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 <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">mTD</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> ) </span> } </span> ;<br/><a NAME="E9:49_1"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> ) </span> } </span>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> ) </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span></span><br/>

<a NAME="E2:49_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> st <br/>( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:49_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E10:49_1"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">TD</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" onmouseover="rs('xxreal_2/D7')" onmouseout="rh()">XXREAL_2:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font><span class="kw"> such that </span><br/><a NAME="E12:49_1"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font>
 <span class="kw">and </span><br/><a NAME="E13:49_1"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ;<br/><a NAME="E14:49_1"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:49_1"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E13:49_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a></span>;<br/><a NAME="E15:49_1"/><span class="lab"><font color="Green" title="E41">A16</font></span>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T12" onmouseover="rs('orders_1/T12')" onmouseout="rh()">ORDERS_1:12</a></span>;<br/><a NAME="E16:49_1"/>
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T25" target="_self" onmouseover="rs('bagorder/T25')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E17:49_1"/><span class="kw">then </span>
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#T4" onmouseover="rs('wellord1/T4')" onmouseout="rh()">WELLORD1:4</a></span>;<br/><a NAME="E18:49_1"/><span class="kw">then </span>
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D5" onmouseover="rs('wellord1/D5')" onmouseout="rh()">WELLORD1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E20:49_1"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> ) </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E21:49_1"/><span class="lab"><font color="Green" title="E43">A18</font></span>: 
<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E9:49_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E14:49_1"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="txt" href="#E15:49_1"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="ref" href="wellord1.html#D3" onmouseover="rs('wellord1/D3')" onmouseout="rh()">WELLORD1:def 3</a></span>;<br/><a NAME="E22:49_1"/><span class="lab"><font color="Green" title="E44">A19</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:49_1"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a></span>;<br/><a NAME="E23:49_1"/><span class="lab"><font color="Green" title="E45">A20</font></span>: 
 ex <font color="Olive" title="b1">a9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> st <br/>( <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Olive" title="b1">a9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:49_1"><span class="lab"><font color="Green" title="E42">A17</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c11">a</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font> )</span><br/><span class="kw">thus </span><a NAME="E24:49_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:49_1"><span class="lab"><font color="Green" title="E45">A20</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font></span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:49_1_3"/><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><a NAME="E4:49_1_3"/><span class="lab"><font color="Green" title="E47">A22</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_3"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:49_1_3"/><span class="lab"><font color="Green" title="E48">A23</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_3"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:49_1_3"/><span class="lab"><font color="Green" title="E49">A24</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1_3"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="ref" href="wellord1.html#T1" onmouseover="rs('wellord1/T1')" onmouseout="rh()">WELLORD1:1</a></span>;<br/><a NAME="E7:49_1_3"/><span class="lab"><font color="Green" title="E50">A25</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c12">a</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="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1_3"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="ref" href="wellord1.html#T1" onmouseover="rs('wellord1/T1')" onmouseout="rh()">WELLORD1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c15">z</font> = <font color="Maroon" title="c13">z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E23:49_1"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="txt" href="#E5:49_1_3"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="ref" href="orders_1.html#T12" onmouseover="rs('orders_1/T12')" onmouseout="rh()">ORDERS_1:12</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:49_1_3_1"/>
(  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c15">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c14">a</font> or  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c14">a</font> or  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c15">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c14">a</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:49_1_3_1_1"/><span class="lab"><font color="Green" title="E51">A26</font></span>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c15">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c14">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:49_1_3_1_1"/>
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c15">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">TD</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1_3"><span class="lab"><font color="Green" title="E48">A23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:49_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:49_1"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="txt" href="#E1:49_1_3_1_1"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="ref" href="xxreal_2.html#D7" onmouseover="rs('xxreal_2/D7')" onmouseout="rh()">XXREAL_2:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="49_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:49_1_3_1_2"/><span class="lab"><font color="Green" title="E51">A27</font></span>: 
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c14">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:49_1_3_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">z</font>,<font color="Maroon" title="c14">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E7:49_1_3"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" onmouseover="rs('bagorder/D7')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E3:49_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A28</font></span>: 
<font color="Maroon" title="c15">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1_3"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="wellord1.html#T1" onmouseover="rs('wellord1/T1')" onmouseout="rh()">WELLORD1:1</a></span>;<br/><a NAME="E4:49_1_3_1_2"/>
<font color="Maroon" title="c15">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font> : ( <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c8">TD</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:49_1"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="txt" href="#E5:49_1_3"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="txt" href="#E1:49_1_3_1_2"><span class="lab"><font color="Green" title="E51">A27</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:49_1_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:49_1"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="txt" href="#E3:49_1_3_1_2"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="49_1_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:49_1_3_1_3"/>
 <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c15">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c14">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:49_1_3_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E7:49_1_3"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" onmouseover="rs('bagorder/D7')" onmouseout="rh()">Def7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E26:49_1"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:49"/><span class="kw">then </span>
 <a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D3" onmouseover="rs('wellord1/D3')" onmouseout="rh()">WELLORD1:def 3</a></span>;<br/>
<a NAME="E10:49"/><span class="kw">then </span>
 <a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="wellord1.html#D5" onmouseover="rs('wellord1/D5')" onmouseout="rh()">WELLORD1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:49"/>
 <a href="bagorder.html#K7" title="BAGORDER:func.7">GrInvLexOrder</a> <font color="Maroon" title="c1">o</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#T4" onmouseover="rs('wellord1/T4')" onmouseout="rh()">WELLORD1:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
