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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b3">I</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span>)</span>  ex <font color="Olive" title="b4">G</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span>)</span> st <font color="Olive" title="b4">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Olive" title="b3">I</font>,<font color="Olive" title="b1">T</font></span><br/><b>let </b><font color="Maroon" title="c2">T</font> be  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b2">I</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span>)</span>  ex <font color="Olive" title="b3">G</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span>)</span> st <font color="Olive" title="b3">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Olive" title="b2">I</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>  ex <font color="Olive" title="b2">G</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> st <font color="Olive" title="b2">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c4">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> st <font color="Olive" title="b1">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font></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" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> or <font color="Maroon" title="c4">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</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" title="E37">A1</font></i>: 
<font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> st <font color="Olive" title="b1">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c5">G</font> = <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>;<br/><b>set </b><font color="Maroon" title="c6">R</font> =  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>;<br/><b>take </b>
<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font></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" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> )</span><br/><b>assume </b><a NAME="E1:50_1_1_1"/><i><font color="Green" title="E38">A2</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font></span><br/><b>then consider </b><font color="Maroon" title="c10">p</font>, <font color="Maroon" title="c11">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_1_1_1"/><i><font color="Green" title="E39">A3</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c11">q</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:50_1_1_1"/>
not <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:50_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">p</font> = <font color="Maroon" title="c10">p</font> as  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E8:50_1_1_1"/>
<font color="Maroon" title="c13">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c12">q</font>,<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_1_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="groeb_1.html#E3:50_1_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">g</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E10:50_1_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></i>;<br/><a NAME="E11:50_1_1_1"/>
<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:50_1_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E12:50_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">p</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:50_1_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></i>;<br/><b>hence </b><a NAME="E13:50_1_1_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5"><i><font color="Green" title="E4">Lm3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E3:50_1_1"/><b>then </b><i><font color="Green" title="E38">A5</font></i>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> is <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="rewrite1.html#D24" title="REWRITE1:def.24">REWRITE1:def 24</a></i>;<br/><a NAME="E4:50_1_1"/>
<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="ideal_1.html#T44" title="IDEAL_1:th.44">IDEAL_1:44</a></i>;<br/><b>hence </b><a NAME="E5:50_1_1"/>
<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="groeb_1.html#D4" target="_self" title="GROEB_1:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><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" title="E37">A6</font></i>: 
<font color="Maroon" title="c4">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> st <font color="Olive" title="b1">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c5">R</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/><b>set </b><font color="Maroon" title="c6">hti</font> =  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">hti</font> =  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #) ;<br/><b>consider </b><font color="Maroon" title="c8">S</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E4:50_1_2"/><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c8">S</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c7">hti</font>, <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="groeb_1.html#T34" target="_self" title="GROEB_1:th.34">Th34</a></i>;<br/><a NAME="E5:50_1_2"/><i><font color="Green" title="E39">A8</font></i>: 
( <font color="Maroon" title="c8">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">hti</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #)  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">hti</font> holds <br/> ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #) st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font> ) ) )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:50_1_2"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="dickson.html#D9" title="DICKSON:def.9">DICKSON:def 9</a></i>;<br/><b>set </b><font color="Maroon" title="c9">M</font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  where <font color="Olive" title="b1">s</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> </span>}</span> ;<br/><a NAME="E6:50_1_2"/>
 ex <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">I</font> st <font color="Olive" title="b1">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">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" title="E40">A9</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">I</font> holds  not <font color="Olive" title="b1">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><i><font color="Green" title="E41">A10</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" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> implies <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:50_1_2_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span></span><br/><a NAME="E2:50_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_2_1"><i><font color="Green" title="E40">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E3:50_1_2_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<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" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> implies <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> )</span><br/><b>assume </b><a NAME="E1:50_1_2_1_2"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font></span><br/><a NAME="E2:50_1_2_1_2"/><b>then </b><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:50_1_2_1_2_1"/><b>then </b>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_2_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="groeb_1.html#E2:50_1_2_1_2"><i><font color="Green" title="E42">A11</font></i></a></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" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:50_1_2_1_2"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="groeb_1.html#E2:50_1_2_1"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c10">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">I</font><b> such that </b><br/><a NAME="E8:50_1_2"/><i><font color="Green" title="E40">A12</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c10">q</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>set </b><font color="Maroon" title="c12">hq</font> =  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c2">T</font>;<br/><b>reconsider </b><font color="Maroon" title="c13">hq</font> =  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c2">T</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #) ;<br/><a NAME="E11:50_1_2"/>
<font color="Maroon" title="c13">hq</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="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E8:50_1_2"><i><font color="Green" title="E40">A12</font></i></a></i>;<br/><a NAME="E12:50_1_2"/><b>then </b><i><font color="Green" title="E41">A13</font></i>: 
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #) st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> &amp; <font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">hq</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:50_1_2"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="dickson.html#D9" title="DICKSON:def.9">DICKSON:def 9</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">S</font>;<br/><a NAME="E14:50_1_2"/>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E12:50_1_2"><i><font color="Green" title="E41">A13</font></i></a></i>;<br/><a NAME="E15:50_1_2"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">s'</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  where <font color="Olive" title="b1">s'</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">s'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> </span>}</span> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c15">M</font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  where <font color="Olive" title="b1">s</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E17:50_1_2"/><i><font color="Green" title="E42">A14</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> holds <br/><font color="Olive" title="b1">x</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> 
 <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" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> implies <font color="Maroon" title="c16">x</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><br/>

<b>assume </b><a NAME="E1:50_1_2_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">x</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><br/>

<b>then consider </b><font color="Maroon" title="c17">s</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:50_1_2_2"/><i><font color="Green" title="E43">A15</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c17">s</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  &amp; <font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> )
 ;<br/>
<a NAME="E4:50_1_2_2"/>
<font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">hti</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="groeb_1.html#E3:50_1_2_2"><i><font color="Green" title="E43">A15</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">q</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E6:50_1_2_2"/><i><font color="Green" title="E44">A16</font></i>: 
( <font color="Maroon" title="c17">s</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; <font color="Maroon" title="c18">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 ;<br/>
<a NAME="E7:50_1_2_2"/>
<font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_2"><i><font color="Green" title="E43">A15</font></i></a>, <a class="txt" href="groeb_1.html#E6:50_1_2_2"><i><font color="Green" title="E44">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:50_1_2_2"/>
<font color="Maroon" title="c16">x</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E18:50_1_2"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">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" title="c16">x</font>, <font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">y</font> implies <font color="Maroon" title="c16">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">y</font> )</span><br/>

<b>assume </b><a NAME="E1:50_1_2_3"/><i><font color="Green" title="E43">A17</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">y</font></span><br/>

<b>then consider </b><font color="Maroon" title="c18">s</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:50_1_2_3"/><i><font color="Green" title="E44">A18</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">s</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  &amp; <font color="Maroon" title="c18">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c19">t</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:50_1_2_3"/><i><font color="Green" title="E45">A19</font></i>: 
( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">t</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  &amp; <font color="Maroon" title="c19">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_2_3"><i><font color="Green" title="E43">A17</font></i></a></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" title="E46">A20</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c20">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">y</font>;<br/><a NAME="E3:50_1_2_3_1"/><i><font color="Green" title="E47">A21</font></i>: 
( <font color="Maroon" title="c20">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c20">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_2_3_1"><i><font color="Green" title="E46">A20</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">r</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E5:50_1_2_3_1"/><i><font color="Green" title="E48">A22</font></i>: 
( <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> &amp; <font color="Maroon" title="c21">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">s</font> &amp; <font color="Maroon" title="c21">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_3"><i><font color="Green" title="E44">A18</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c22">v</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E7:50_1_2_3_1"/><i><font color="Green" title="E49">A23</font></i>: 
( <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">v</font> &amp; <font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c22">v</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">t</font> &amp; <font color="Maroon" title="c22">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2_3"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="groeb_1.html#E3:50_1_2_3_1"><i><font color="Green" title="E47">A21</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#E1:50_1_2_3"><i><font color="Green" title="E43">A17</font></i></a>, <a class="txt" href="groeb_1.html#E3:50_1_2_3"><i><font color="Green" title="E44">A18</font></i></a>, <a class="txt" href="groeb_1.html#E5:50_1_2_3"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="groeb_1.html#E5:50_1_2_3_1"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="groeb_1.html#E7:50_1_2_3_1"><i><font color="Green" title="E49">A23</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:50_1_2_3"/>
<font color="Maroon" title="c16">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c16">G'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:50_1_2"/><i><font color="Green" title="E43">A24</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c16">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E17:50_1_2"><i><font color="Green" title="E42">A14</font></i></a>, <a class="ref" href="wellord2.html#T27" title="WELLORD2:th.27">WELLORD2:27</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">xx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">M</font>;<br/><b>consider </b><font color="Maroon" title="c18">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E23:50_1_2"/><i><font color="Green" title="E44">A25</font></i>: 
<font color="Maroon" title="c16">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E20:50_1_2"><i><font color="Green" title="E43">A24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">G'</font> = <font color="Maroon" title="c16">G'</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="groeb_1.html#E23:50_1_2"><i><font color="Green" title="E44">A25</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c20">G</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">G'</font> :  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c21">xx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">M</font>;<br/><b>consider </b><font color="Maroon" title="c22">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E27:50_1_2"/><i><font color="Green" title="E45">A26</font></i>: 
<font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c21">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c22">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E20:50_1_2"><i><font color="Green" title="E43">A24</font></i></a></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" title="c23">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">G'</font> :  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  implies <font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:50_1_2_4"/>
<font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">G'</font> :  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c24">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">G'</font><b> such that </b><br/><a NAME="E3:50_1_2_4"/><i><font color="Green" title="E46">A27</font></i>: 
( <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x</font> &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c24">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c25">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:50_1_2_4"/><i><font color="Green" title="E47">A28</font></i>: 
( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c24">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_4"><i><font color="Green" title="E46">A27</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c26">s</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E7:50_1_2_4"/><i><font color="Green" title="E48">A29</font></i>: 
( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">s</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  &amp; <font color="Maroon" title="c26">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2_4"><i><font color="Green" title="E47">A28</font></i></a></i>;<br/><a NAME="E8:50_1_2_4"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">y</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2_4"><i><font color="Green" title="E47">A28</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E9:50_1_2_4"/><b>then </b>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c27">q</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E11:50_1_2_4"/><i><font color="Green" title="E49">A30</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">q</font> &amp; <font color="Maroon" title="c27">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c27">q</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">s</font> &amp; <font color="Maroon" title="c27">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:50_1_2_4"><i><font color="Green" title="E48">A29</font></i></a></i>;<br/><b>thus </b><a NAME="E12:50_1_2_4"/>
<font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_4"><i><font color="Green" title="E46">A27</font></i></a>, <a class="txt" href="groeb_1.html#E11:50_1_2_4"><i><font color="Green" title="E49">A30</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c23">G</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">G'</font> :  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E30:50_1_2"/><i><font color="Green" title="E46">A31</font></i>: 
<font color="Maroon" title="c15">M</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">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" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span> ;<br/>
<a NAME="E1:50_1_2_5"/><i><font color="Green" title="E47">A33</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c24">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:50_1_2_5"/><i><font color="Green" title="E48">A34</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">S</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="groeb_1.html#E1:50_1_2_5"><i><font color="Green" title="E47">A33</font></i></a>);<br/></i>
<div><i><font color="Green" title="E49">A35</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" title="c25">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> implies <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font> )</span><br/><b>assume </b><a NAME="E1:50_1_2_5_1"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font></span><br/><b>then consider </b><font color="Maroon" title="c26">s</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:50_1_2_5_1"/><i><font color="Green" title="E50">A36</font></i>: 
( <font color="Maroon" title="c25">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">s</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  &amp; <font color="Maroon" title="c26">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> )
 ;<br/><a NAME="E4:50_1_2_5_1"/>
<font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_5"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="groeb_1.html#E3:50_1_2_5_1"><i><font color="Green" title="E50">A36</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E5:50_1_2_5_1"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_5"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="groeb_1.html#E3:50_1_2_5_1"><i><font color="Green" title="E50">A36</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<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" title="c25">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font> implies <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> )</span><br/><b>assume </b><a NAME="E1:50_1_2_5_2"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font></span><br/><b>then consider </b><font color="Maroon" title="c26">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_1_2_5_2"/><i><font color="Green" title="E50">A37</font></i>: 
( <font color="Maroon" title="c26">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">f</font> &amp; <font color="Maroon" title="c25">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">s</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:50_1_2_5_2"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">s</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_5"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="groeb_1.html#E3:50_1_2_5_2"><i><font color="Green" title="E50">A37</font></i></a></i>;<br/><b>hence </b><a NAME="E5:50_1_2_5_2"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_5"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="groeb_1.html#E3:50_1_2_5_2"><i><font color="Green" title="E50">A37</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:50_1_2_5"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">M</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E4:50_1_2_5"><i><font color="Green" title="E49">A35</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E7:50_1_2_5"/>
<font color="Maroon" title="c15">M</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_5"><i><font color="Green" title="E48">A34</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">$2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">G</font> );<br/><a NAME="E31:50_1_2"/><i><font color="Green" title="E47">A39</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c24">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>assume </b><a NAME="E1:50_1_2_6"/><i><font color="Green" title="E48">A40</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c24">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<b>then consider </b><font color="Maroon" title="c25">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_1_2_6"/><i><font color="Green" title="E49">A41</font></i>: 
<font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c25">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E20:50_1_2"><i><font color="Green" title="E43">A24</font></i></a></i>;<br/>
<a NAME="E4:50_1_2_6"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c24">x</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_6"><i><font color="Green" title="E49">A41</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">y</font> = <font color="Maroon" title="c25">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">G'</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:50_1_2_6"/>
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">G</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:50_1_2_6"><i><font color="Green" title="E48">A40</font></i></a>, <a class="txt" href="groeb_1.html#E3:50_1_2_6"><i><font color="Green" title="E49">A41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:50_1_2_6"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c24">x</font>,<font color="Olive" title="b1">y</font>]
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_6"><i><font color="Green" title="E49">A41</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c24">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E33:50_1_2"/><i><font color="Green" title="E48">A42</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">M</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="groeb_1.html#E31:50_1_2"><i><font color="Green" title="E47">A39</font></i></a>);<br/></i><div><i><font color="Green" title="E49">A43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">G</font> implies <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font> )</span><br/><b>assume </b><a NAME="E1:50_1_2_7"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font></span><br/><b>then consider </b><font color="Maroon" title="c26">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">G'</font><b> such that </b><br/><a NAME="E3:50_1_2_7"/><i><font color="Green" title="E50">A44</font></i>: 
( <font color="Maroon" title="c25">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x</font> &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c26">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c27">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:50_1_2_7"/><i><font color="Green" title="E51">A45</font></i>: 
( <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c26">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_7"><i><font color="Green" title="E50">A44</font></i></a></i>;<br/><a NAME="E6:50_1_2_7"/><i><font color="Green" title="E52">A46</font></i>: 
<font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E33:50_1_2"><i><font color="Green" title="E48">A42</font></i></a>, <a class="txt" href="groeb_1.html#E5:50_1_2_7"><i><font color="Green" title="E51">A45</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E7:50_1_2_7"/>
( <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">G</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E33:50_1_2"><i><font color="Green" title="E48">A42</font></i></a>, <a class="txt" href="groeb_1.html#E5:50_1_2_7"><i><font color="Green" title="E51">A45</font></i></a></i>;<br/><a NAME="E8:50_1_2_7"/><b>then </b>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2_7"><i><font color="Green" title="E51">A45</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E9:50_1_2_7"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_7"><i><font color="Green" title="E50">A44</font></i></a>, <a class="txt" href="groeb_1.html#E6:50_1_2_7"><i><font color="Green" title="E52">A46</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><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" title="c25">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font> implies <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">G</font> )</span><br/><b>assume </b><a NAME="E1:50_1_2_8"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">G</font></span><br/><b>then consider </b><font color="Maroon" title="c26">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_1_2_8"/><i><font color="Green" title="E50">A47</font></i>: 
( <font color="Maroon" title="c26">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">f</font> &amp; <font color="Maroon" title="c25">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">s</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>thus </b><a NAME="E4:50_1_2_8"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">G</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E33:50_1_2"><i><font color="Green" title="E48">A42</font></i></a>, <a class="txt" href="groeb_1.html#E3:50_1_2_8"><i><font color="Green" title="E50">A47</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E36:50_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">G</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E34:50_1_2"><i><font color="Green" title="E49">A43</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c25">G</font> = <font color="Maroon" title="c23">G</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="txt" href="groeb_1.html#E27:50_1_2"><i><font color="Green" title="E45">A26</font></i></a>, <a class="txt" href="groeb_1.html#E30:50_1_2"><i><font color="Green" title="E46">A31</font></i></a>, <a class="txt" href="groeb_1.html#E33:50_1_2"><i><font color="Green" title="E48">A42</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/><b>take </b>
<font color="Maroon" title="c25">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c25">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font></span><br/><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" title="c26">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">G</font> implies <font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> )</span><br/><b>assume </b><a NAME="E1:50_1_2_9"/>
<font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font></span><br/><b>then consider </b><font color="Maroon" title="c27">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">G'</font><b> such that </b><br/><a NAME="E3:50_1_2_9"/><i><font color="Green" title="E50">A48</font></i>: 
( <font color="Maroon" title="c26">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">x</font> &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c27">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c28">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:50_1_2_9"/><i><font color="Green" title="E51">A49</font></i>: 
( <font color="Maroon" title="c28">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font> &amp; <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c28">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c27">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_9"><i><font color="Green" title="E50">A48</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c29">s</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E7:50_1_2_9"/><i><font color="Green" title="E52">A50</font></i>: 
( <font color="Maroon" title="c28">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c29">s</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  &amp; <font color="Maroon" title="c29">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2_9"><i><font color="Green" title="E51">A49</font></i></a></i>;<br/><a NAME="E8:50_1_2_9"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c28">y</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:50_1_2_9"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E9:50_1_2_9"/><b>then </b>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">q</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E11:50_1_2_9"/><i><font color="Green" title="E53">A51</font></i>: 
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> &amp; <font color="Maroon" title="c30">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c30">q</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c29">s</font> &amp; <font color="Maroon" title="c30">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:50_1_2_9"><i><font color="Green" title="E52">A50</font></i></a></i>;<br/><b>thus </b><a NAME="E12:50_1_2_9"/>
<font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:50_1_2_9"><i><font color="Green" title="E50">A48</font></i></a>, <a class="txt" href="groeb_1.html#E11:50_1_2_9"><i><font color="Green" title="E53">A51</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E39:50_1_2"/><b>then </b><i><font color="Green" title="E50">A52</font></i>: 
<font color="Maroon" title="c25">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E40:50_1_2"/>
for <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font> holds <br/> ex <font color="Olive" title="b2">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c25">G</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b2">b'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b1">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font> implies  ex <font color="Olive" title="b1">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c25">G</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b1">b'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c26">b</font> ) )</span><br/>

<b>assume </b><a NAME="E1:50_1_2_10"/><i><font color="Green" title="E51">A53</font></i>: 
<font color="Maroon" title="c26">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c25">G</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b1">b'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c26">b</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c27">bb</font> = <font color="Maroon" title="c26">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #) <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">bb'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #)<b> such that </b><br/><a NAME="E4:50_1_2_10"/><i><font color="Green" title="E52">A54</font></i>: 
( <font color="Maroon" title="c28">bb'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> &amp; <font color="Maroon" title="c28">bb'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c27">bb</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:50_1_2"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="groeb_1.html#E1:50_1_2_10"><i><font color="Green" title="E51">A53</font></i></a>, <a class="ref" href="dickson.html#D9" title="DICKSON:def.9">DICKSON:def 9</a></i>;<br/>
<a NAME="E5:50_1_2_10"/>
<font color="Maroon" title="c28">bb'</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c29">b'</font> = <font color="Maroon" title="c28">bb'</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c29">b'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c29">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c25">G</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c29">b'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c26">b</font> )</span><br/>

<a NAME="E7:50_1_2_10"/><i><font color="Green" title="E53">A55</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">bb'</font>,<font color="Maroon" title="c27">bb</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="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E4:50_1_2_10"><i><font color="Green" title="E52">A54</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c30">N</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">bb'</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span> ;<br/>
<a NAME="E8:50_1_2_10"/><i><font color="Green" title="E54">A56</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">bb'</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E4:50_1_2_10"><i><font color="Green" title="E52">A54</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c31">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:50_1_2_10"/><i><font color="Green" title="E55">A57</font></i>: 
<font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">bb'</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c31">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E20:50_1_2"><i><font color="Green" title="E43">A24</font></i></a></i>;<br/>
<a NAME="E11:50_1_2_10"/><i><font color="Green" title="E56">A58</font></i>: 
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">G'</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">bb'</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="groeb_1.html#E10:50_1_2_10"><i><font color="Green" title="E55">A57</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c32">y</font> = <font color="Maroon" title="c31">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">G'</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:50_1_2_10"/>
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">bb'</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="groeb_1.html#E11:50_1_2_10"><i><font color="Green" title="E56">A58</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c33">r</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E15:50_1_2_10"/><i><font color="Green" title="E57">A59</font></i>: 
( <font color="Maroon" title="c32">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">r</font> &amp; <font color="Maroon" title="c33">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c33">r</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">bb'</font> &amp; <font color="Maroon" title="c33">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 ;<br/>
<a NAME="E16:50_1_2_10"/>
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">G</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E8:50_1_2_10"><i><font color="Green" title="E54">A56</font></i></a>, <a class="txt" href="groeb_1.html#E10:50_1_2_10"><i><font color="Green" title="E55">A57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:50_1_2_10"/>
( <font color="Maroon" title="c29">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c25">G</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c29">b'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c26">b</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:50_1_2_10"><i><font color="Green" title="E53">A55</font></i></a>, <a class="txt" href="groeb_1.html#E15:50_1_2_10"><i><font color="Green" title="E57">A59</font></i></a>, <a class="ref" href="groeb_1.html#D5" target="_self" title="GROEB_1:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E41:50_1_2"/><b>then </b>
 <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="groeb_1.html#K3" title="GROEB_1:func.3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c25">G</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="groeb_1.html#T28" target="_self" title="GROEB_1:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E42:50_1_2"/>
<font color="Maroon" title="c25">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E39:50_1_2"><i><font color="Green" title="E50">A52</font></i></a>, <a class="ref" href="groeb_1.html#T29" target="_self" title="GROEB_1:th.29">Th29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
