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<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#V2">admissible</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V1">add-closed</a> <a href="ideal_1.html#V2">left-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> or <font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">G</font> = <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>;<br/><b>set </b><font color="Maroon">R</font> =  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">T</font>;<br/><b>take </b>
<span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_1_1_1"/><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">T</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">T</font> )
 ;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:50_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:50_1_1_1"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:50_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E9:50_1_1_1"/>
<font color="Maroon">p</font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">q</font>,<span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">T</font>
 <b>by </b><i>, , <a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:50_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">p</font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">q</font>,<font color="Maroon">g</font>,<font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E12:50_1_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:50_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="polyred.html#R6">is_reducible_wrt</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><b>hence </b><a NAME="E14:50_1_1_1"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="rewrite1.html#R5">are_convergent_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:50_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">T</font> is <a href="rewrite1.html#V9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="rewrite1.html#D24">REWRITE1:def 24</a></i>;<br/><a NAME="E4:50_1_1"/>
<span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T44">IDEAL_1:44</a></i>;<br/><b>hence </b><a NAME="E5:50_1_1"/>
<span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> <a href="groeb_1.html#R2" target="_self">is_Groebner_basis_of</a> <font color="Maroon">I</font>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><a NAME="E1:50_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">R</font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #);<br/><b>set </b><font color="Maroon">hti</font> =  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">I</font>,<font color="Maroon">T</font>;<br/><b>reconsider </b><font color="Maroon">hti</font> =  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">I</font>,<font color="Maroon">T</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #) ;<br/><b>consider </b><font color="Maroon">S</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E4:50_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S</font> <a href="dickson.html#R1">is_Dickson-basis_of</a> <font color="Maroon">hti</font>, <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #)
 <b>by </b><i/>;<br/><a NAME="E5:50_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hti</font> &amp; ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #)  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">hti</font> holds <br/> ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #) st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> ) ) )
 <b>by </b><i>, <a class="ref" href="dickson.html#D9">DICKSON:def 9</a></i>;<br/><b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Olive">s</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  where <font color="Olive">s</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> </span>}</span> ;<br/><a NAME="E6:50_1_2"/>
 ex <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> st <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_2_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> holds  not <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/>

<div><i><font color="Green">E73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_1_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:50_1_2_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:50_1_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_1_2_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 ;<br/><a NAME="E2:50_1_2_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2_1_2_1"/>
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:50_1_2_1_2_1"/><b>then </b>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:50_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:50_1_2_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E8:50_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>set </b><font color="Maroon">hq</font> =  <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>;<br/><b>reconsider </b><font color="Maroon">hq</font> =  <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #) ;<br/><a NAME="E11:50_1_2"/>
<font color="Maroon">hq</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font></span>)</span> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E12:50_1_2"/><b>then </b>
<font color="Maroon">hq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">hti</font>
 ;<br/><a NAME="E13:50_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #) st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">hq</font> )
 <b>by </b><i>, <a class="ref" href="dickson.html#D9">DICKSON:def 9</a></i>;<br/><b>consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><a NAME="E15:50_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E16:50_1_2"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">r</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Olive">s'</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  where <font color="Olive">s'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">s'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> </span>}</span> 
 ;<br/><b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Olive">s</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  where <font color="Olive">s</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E18:50_1_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:50_1_2_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/>
<a NAME="E4:50_1_2_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">hti</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:50_1_2_2"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font></span>)</span> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:50_1_2_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">s</font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 ;<br/>
<a NAME="E8:50_1_2_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:50_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E19:50_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_2_3"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:50_1_2_3"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/>
<b>consider </b><font color="Maroon">t</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E5:50_1_2_3"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Maroon">t</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2_3_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font>;<br/><a NAME="E3:50_1_2_3_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:50_1_2_3_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">r</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">v</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:50_1_2_3_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">v</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 <b>by </b><i>, <a class="txt" href="groeb_1.html#E1:50_1_2"><i><font color="Green">E47</font></i></a></i>;<br/><b>thus </b><a NAME="E8:50_1_2_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="groeb_1.html#E4:50_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:50_1_2_3"/>
<font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">G'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:50_1_2"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="wellord2.html#T27">WELLORD2:27</a></i>;<br/><b>consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:50_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G'</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/><b>set </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G'</font> :  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E28:50_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_1_2_4"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G'</font> :  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G'</font><b> such that </b><br/><a NAME="E3:50_1_2_4"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:50_1_2_4"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">s</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E7:50_1_2_4"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_2"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E8:50_1_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:50_1_2_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:50_1_2_4"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:50_1_2"><i><font color="Green">E48</font></i></a></i>;<br/><b>thus </b><a NAME="E12:50_1_2_4"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G'</font> :  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E31:50_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">M</font> is <a href="finset_1.html#V1">finite</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_5"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span> ;<br/>
<a NAME="E1:50_1_2_5"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E2:50_1_2_5"/><i><font color="Green">E163</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:50_1_2_5"/><i><font color="Green">E164</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="groeb_1.html#E1:50_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="groeb_1.html#E4:50_1_2"><i><font color="Green">E48</font></i></a>);<br/></i>
<div><i><font color="Green">E165</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_1_2_5_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:50_1_2_5_1"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/><a NAME="E4:50_1_2_5_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E5:50_1_2_5_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_1_2_5_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_1_2_5_2"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:50_1_2_5_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a></i>;<br/><b>hence </b><a NAME="E5:50_1_2_5_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:50_1_2_5"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E8:50_1_2_5"/>
<font color="Maroon">M</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> );<br/><a NAME="E32:50_1_2"/><i><font color="Green">E168</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_2_6"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y2</font>] )
 ;<br/>

<a NAME="E2:50_1_2_6"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:50_1_2_6"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E33:50_1_2"/><i><font color="Green">E169</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_2_7"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_1_2_7"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E4:50_1_2_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G'</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:50_1_2_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E7:50_1_2_7"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E35:50_1_2"/><i><font color="Green">E172</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="groeb_1.html#E1:50_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="groeb_1.html#E4:50_1_2"><i><font color="Green">E48</font></i></a>);<br/></i><div><i><font color="Green">E173</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_1_2_8"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G'</font><b> such that </b><br/><a NAME="E3:50_1_2_8"/><i><font color="Green">E174</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:50_1_2_8"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a></i>;<br/><a NAME="E6:50_1_2_8"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E7:50_1_2_8"/>
( <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E8:50_1_2_8"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E9:50_1_2_8"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_1_2_9"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_1_2_9"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>thus </b><a NAME="E4:50_1_2_9"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E38:50_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i>, , <a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/><b>take </b>
<font color="Maroon">G</font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_1_2_10"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G'</font><b> such that </b><br/><a NAME="E3:50_1_2_10"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:50_1_2_10"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a></i>;<br/><b>consider </b><font color="Maroon">s</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E7:50_1_2_10"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/><a NAME="E8:50_1_2_10"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:50_1_2_10"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:50_1_2_10"/><i><font color="Green">E181</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E12:50_1_2_10"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E41:50_1_2"/><b>then </b><i><font color="Green">E182</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E42:50_1_2"/>
for <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">I</font>,<font color="Maroon">T</font> holds <br/> ex <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> &amp; <font color="Olive">b'</font> <a href="polynom1.html#R3">divides</a> <font color="Olive">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>

<b>assume </b><a NAME="E1:50_1_2_11"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">I</font>,<font color="Maroon">T</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #) <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">bb'</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #)<b> such that </b><br/><a NAME="E4:50_1_2_11"/><i><font color="Green">E184</font></i>: 
( <font color="Maroon">bb'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">bb'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">bb</font> )
 <b>by </b><i>, , <a class="ref" href="dickson.html#D9">DICKSON:def 9</a></i>;<br/>
<a NAME="E5:50_1_2_11"/>
<font color="Maroon">bb'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">bb'</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> ;<br/>
<b>take </b>
<font color="Maroon">b'</font>
;<br/>

<a NAME="E7:50_1_2_11"/><i><font color="Green">E185</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">bb'</font>,<font color="Maroon">bb</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">N</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb'</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span> ;<br/>
<a NAME="E8:50_1_2_11"/><i><font color="Green">E186</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb'</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:50_1_2_11"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb'</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E11:50_1_2_11"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb'</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G'</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:50_1_2_11"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb'</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="groeb_1.html#E4"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E15:50_1_2_11"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">r</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb'</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 ;<br/>
<a NAME="E16:50_1_2_11"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:50_1_2_11"/><b>then </b>
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font></span>)</span> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E18:50_1_2_11"/>
( <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">b'</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="groeb_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E43:50_1_2"/><b>then </b>
 <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">I</font>,<font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="groeb_1.html#K3" target="_self">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">G</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E44:50_1_2"/>
<font color="Maroon">G</font> <a href="groeb_1.html#R2" target="_self">is_Groebner_basis_of</a> <font color="Maroon">I</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
